Only One Me Allowed Right Now

In most time travel/cloning/alternate reality stories, one character can have multiple copies of themselves running around in the same time period. Some might have problems if those copies meet each other. Then there are these cases...

Only One Me Allowed Right Now is a case where the universe either flat out denies multiple copies of a character to exist in a same time period, or that either the character and the copies go crazy or the universe starts to break down, or something bad happens.

Note that this is NOT Never the Selves Shall Meet. In that one, you can have millions of copies of a same character running around in the same universe without that much trouble, only they must not meet each other. In this case, even though there are only 2 copies and they are at the opposite side of the universe, the problem still happens.

Anime and Manga

 * In one episode of Sgt Frog, Keroro accidentally clones himself a thousandfold using the Kero Ball. After various hijinks, the clones start to fade. Kululu gives a Techno Babble explanation, but when told by Natsumi to "use words real people understand," he gets down to the point: if they don't destroy all the clones, then the Kero Ball will overload and all the Keroros, including the original, will disappear.
 * This turns out to be a major plot point in XXX Holic;  is a time travel duplicate. As a result, he hates himself and draws in supernatural beings trying to grant his wish and kill him, and if he doesn't develop strong social connections he'll soon cease to exist as reality corrects itself.

Comic Books

 * This applied to the DC Universe before the first Crisis. If a character traveled to a time where he or she already existed, it (the version not belonging to that time) would become an invisible, unheard, ineffectual phantom until it stepped out of that moment in time. Note that back then, even if characters were allowed to "meet themselves" history could not be changed, so it was pointless anyway.
 * Superman's childhood friend Pete Ross, of all people, found a way around this. He was furious at Superman at the time (he blamed Superman for his son getting kidnapped by aliens; long story), and wanted a way to fight him. So he got ahold of some phlebotinum that let him swap minds with Supes's younger self, Superboy, and then returned to the present in Superboy's body, to duke it out with adult Supes. It turns out the Only One Me rule only applies to your mind being in two places at once, not your body. This actually makes a degree of sense, since even if you travel back to before your birth or after your death, the matter that makes up your body should still be around somewhere, and yet you only appear as a phantom iif you travel within your own lifetime.
 * Some early Silver Age Superman stories (& one Justice League tale) used the idea that if Superman traveled within his own lifetime the earlier version would take his place in the present. So if Superman traveled to when he was Superbaby, Superbaby would appear in the present while Superman was in the past.
 * In Nth Man the Ultimate Ninja, it is impossible for a person to be in two places at once. This causes one character to spontaneously combust when she returns to the past, as she arrives at the same time she's being born.
 * This happened to Jubilee in an issue of Wolverine: In her youth, there was an incident where she was in a car with her friends, who suddenly asked her why she momentarily disappeared into thin air. Not remembering doing so, she dismisses it as her friends acting crazy. Years later, she briefly falls into the time portal belonging to Gateway, appearing in her parents' house, at the exact same time she "disappeared" in her friends' car. It is explained that two of her couldn't exist in the same point in time, so her younger self simply vanished until the older version returned to the present.

Film

 * In the French film Les Visiteurs, an object near a copy of itself (from earlier or later in the time stream) will try to merge with its past and/or future selves. Violently. "Near" isn't precisely defined, but seems to vaguely obey the inverse square law.

Literature

 * In Star Wars Expanded Universe, this is believed to be the case with clone madness when the clone is created if the process is accelerated, like using Spaarti cloning cylinder.
 * To be more specific, it relates to the individual Force-signatures of their minds. Clones have identical Force-signatures, and this exerts pressure on their minds as they develop, even if they aren't otherwise Force-sensitive. If they are grown any faster than double-speed, their minds can't adjust to the strain, and break. So a clone army would take about ten years to grow under ideal conditions.
 * In Connie Willis's Blackout / All Clear, a time traveler has to return before other times that he visited -- his arrival then is his "deadline". An important plot point.
 * Earlier, in To Say Nothing of the Dog, this was a minor point. One character could be sent back a few days because during those days, they were unable to pick him up from his time travel.
 * As that universe is a Stable Time Loop where paradoxes are stopped by the universe disallowing the trip from happening at all, a more interesting way of looking at To Say Nothing of the Dog is that retrieving him from the past would have killed his near-future self, which the police would have identified as him, cause a paradox. So it refused to let his past self through.
 * In Dean Koontz's Lightning, the inventors of time traveling discover that the universe has a built-in anti-paradox mechanism, where you simply get bounced back from the time-gate if you are attempting to travel to a time where you're already present (or even might be - one character tries to correct a mistake by traveling to a time a couple of minutes before he last showed up, and the universe doesn't let him).
 * In the Time Scout series, you can't travel to a time where one of you already exists because you'll wink out of existence on arrival.
 * In the series Dragonriders of Pern, going back in time to a period where you were still alive causes both versions of yourself to experience physical and mental distress. It gets worse the two selves are near each other. In the first novel, for example, F'Nor spent some five years both carrying out his normal duties in Benden Weyr, and setting up a new Weyr in the Southern Continent. They were generally alright when on opposite ends of the planet, but whenever Future!F'Nor came to Benden Weyr to tell F'lar something, the proximity caused great duress.

Live Action TV

 * In Seven Days when Parker goes back in time, the Parker from the time he goes back to disappears from existence (as does the time machine itself and anything else inside it).
 * This causes a problem once, as Parker taking back something causes it to impossibly disappear out of a locked-and-handcuffed-to-a-person briefcase of the villain, cluing him in that something very very strange was going on.
 * There was also an episode where Parker spilling tea on a console resulted in the Sphere jumping prematurely (i.e. before he got in). So when the alarm sounds seven days ago, the others are very surprised to see their versions of Parker and Donovan (the backup chrononaut) still around. All they have to go on are the contents of Parker's bag from the future, including a key. Luckily, Dr. Mentnor knows a genuine psychic.
 * Not time-travel related, but early in Stargate SG-1, when two of the same character from alternate realities met up with one another, there would be negative "feedback" that would kill them if they stayed in the same reality too long.
 * Of course, conveniently, when the alternate Sam, Jack, and Teal'c arrive to Ancient Egypt in "Moebius", the original Daniel explains that their counterparts were killed by the Goa'uld. When he asks about his version, they reveals that Teal'c killed him after that Daniel was implanted with a symbiote.
 * Another convenient example in "The Road Not Taken" has Sam accidentally end up in a parallel world while experimenting with phase-shifting technology behind a force field, while her double from that universe was doing something similar. The accident that threw "our" Sam into the other universe also killed her double.
 * This is completely ignored in later episodes featuring multiple versions of the same characters. Namely, an episode dozens of SG-1s are showing up and stay there for several days with no side effects.
 * In Mirror Mirror, things that exist at both times are prevented from crossing the mirror: If you try to take one with you, you'll be shocked instead.

Tabletop RPG's

 * Dungeons and Dragons 1st and 2nd edition. When a clone was created with the Clone spell, both the original and the clone knew of the other's existence and each would try to kill the other. If they couldn't, within a week either one would go insane and kill itself (90% likely to be the clone) or, 2% of the time, both would kill themselves.
 * In Forgotten Realms some mages found a way around this -- Zunroun cooperated with a dozen of his clones for some time. Preventing this by keeping all clones cold until the original dies also worked -- Manshoon's Stasis Clone spell was a great secret and major plot point, making the Black Network's leader a Recurring Boss while keeping him on a disproportionately low level due to setbacks and memory losses every time he bit off too much to chew and got killed.
 * The Gates Of Hell, a fan written Dungeons and Dragons book, features a devil who can travel in time. If he tries coming too close to his alternate self, he is pushed back and stunned.

Video Games

 * In Infinite Space, a rogue Zenito general repeatedly used mind transfer to escape death when you killed him.
 * When the main characters of the War Craft Expanded Universe War of the Ancients Trilogy are transported to the past, Krasus, the only one who existed then, finds himself considerably weaker and unable to transform into his true dragon form of Korialstrasz. We find Krasus past self is experiencing the same problems, and it's attributed to the fact that, since they're one and the same, they're sharing the same life force.
 * They find themselves much stronger if they are together.
 * Castlevania Judgment is a borderline example. While Aeon is able to face himself in battle, he specifically states that it will cause damage to the timeline. Since he has the power to fix it, however, there isn't much of an issue.
 * In the Mirror Realm of Adventure Quest Worlds's 2nd Birthday Event, the hero that you play is chosen to save the Mirror Realm because he or she is the only person in the entire multiverse who does not have an alternate in the Mirror Realm -- if anyone else in either world were to go to the other world, they'd have to switch places with their mirror counterpart. Paul and Storm do not have mirror counterparts either because they "grew up in a small suburb just outside of the multiverse." Just roll with it.
 * In the The Legend of Zelda: Majoras Mask, no matter how many times you go back in time, there is only ever one Link (and Tatl) in Termina. Most likely, this is the Ocarina of Time at work.

Web Comics

 * Homestuck: According to Aradia, a future self who travels back from a doomed timeline is doomed to be destroyed by the universe sooner or later.
 * This is only a partial example, as people can travel through time without problems as long as they stay in the same universe.

Western Animation

 * In Justice League, Vandal Savage's time machine cannot send a person back to when that person existed.
 * Futurama: Bender's Big Score had this in effect. When ever a duplicate was created through time travel the universe would kill them off to prevent further confusion. However, at the end of the movie, several hundred time-traveling Benders appear simultaneously; this proves too much for the universe to handle, and a hole in space-time rips open.
 * In Oggy and The Cockroaches, characters traveling to the past "merge" with their past selves upon meeting to avoid this problem.

Real Life

 * Averted big time. According to quantum physics, there is only one of each particle. That is, if you pick two electrons at random, they are the same electron. This electron exists in about 10^80 places at once. Time travel isn't even necessary.