Paused Interrupt

Sometimes, the dialog isn't timed perfectly. In other words, when one character interrupts another character, it isn't likely to sound like an actual interruption, but merely that one character finished a line which ended in mid sentence and the other character began their own line. Example:

"Rosh: Hey, it was just a joke! I didn't think it would ... [beat] Kyle: If that droid had been set to Luke's training regimen, it would have killed Jaden! What were you thinking?"

Most likely in video games, in which the individual lines of dialog are stored as separate sound files, and the actors themselves—no matter what their quality is—cannot correct this, and (for some reason, possibly bad programming) neither can the game engine. There's also the fact that, in many cases, VO studios have to receive line readings through correspondence. It would be too difficult to gather each actor in the studio to record simultaneously. Also occurs in relatively amateur productions, or in relatively amateur English translations.

In games where you need to press a button to advance the text, this is more or less guaranteed to happen.

Note how this is primarily a voice-acting trope. While it may show up in some text-dialogue games, it is quite feasible to show with word balloons one character interrupting another—while character A's dialogue is appearing, have character B's balloon open up over A's balloon (and speech) and start up.

Film
"Ferro: Move it, Spunkmeyer. We're rollin’. Spunkmeyer: Hold on a second. There's something... [Unnatural sounding pause] Ferro: Just get up here!"
 * Notably averted in the films of Robert Altman, which are famous for their overlapping dialogue. This was lampshaded by Lily Tomlin and Meryl Streep as they repeatedly talked over each other while presenting Altman's Lifetime Achievement Oscar.
 * Most of the dialogue and performances in Aliens are very good, which it glaring when one of the characters stops their line dead for no reason before being interrupted:

Live Action TV

 * Garth Marenghi's Darkplace is crammed with deliberately bad acting and includes at least one example of this trope.
 * In the Stargate Atlantis episode The Eye, Weir's conversation with Sheppard was supposed to be interrupted by Kolya yanking the radio away. She stopped her line five seconds before it moved.
 * Happens at times in the 1995 BBC version of Pride and Prejudice, where you can just hear that there really wasn't anything else in that line, even though it stops mid-sentence.
 * Happens in The West Wing. The rapid-fire speech of the characters frequently involves one character interrupting another. All to the good, but the scripts always ended an interrupted character's line on a whole word, making this last word the cue for the interrupting character to take over. The end result is something similar to Fred and George's alternating speech than a genuine interruption.

Video Games
""You're comparing yourself to me? Ha! You're not even good enough to b--" "I'll make you eat those words!" "--e my fake!""
 * The sample quote came from one of the Dark Forces games.
 * In Grand Theft Auto: San Andreas there is a segment where an NPC, Catalina, calls the protagonist up on his cell phone to chew him out. As with most phone conversations in the game, this happens in real time. She flies into her usual rage, and hangs up on the player—but the game doesn't know how to handle a dramatic mid-call hangup. She just stops talking, there is no click or dial tone, and the protagonist calmly folds up his phone and puts it away without a second glance, precisely the same way he is animated to end every (complete) phone conversation.
 * Seeing as how Catalina is insane, the animation can be handwaved as CJ not caring.
 * Also can be averted by the player, who can cancel a phone call at any time without any paused interrupts.
 * At the beginning of the first Resident Evil, a particularly long pause can be found when a conversation is interrupted by a gunshot.
 * Inversion: Sonic Adventure 2's localization took the rough (read: almost Engrish) translations of the Japanese script and placed them in the same sized sound files. Thus, the characters speak at the same time frequently, or may say things that don't make a bit of sense. The most hilarious example has one character repeating part of a line in a news report before it's even said.

""Heavy weapon! Get down!" [pause] "Oh my god! What do we do?!?'' [pause] "Do you think they saw us?" [pause] [gunfire]"
 * Full Spectrum Warrior for the PlayStation 2 had notoriously long load times between lines of dialogue. The resulting irregular timing completely disrupted several otherwise tense scenes.

"Necron: My Lord, the living have... [Pause] [BOOM!]"
 * Deus Ex was bad for this.
 * XIII had a scene near the beginning of the game in which this happens several times, not through dialogue, but through sound effects. As you're being interrogated, Amos is explaining to you his findings when he's interrupted by his intercom beeping - he stops talking seconds later. Later in the same scene, he's interrupted by a VERY LOUD EXPLOSION, yet keeps talking for almost three whole seconds before finally stopping to yell "What the devil's going on here?!"
 * This applies to all kinds of conversations in the game...making mandatory, unskippable scenes particularly painful to watch if you have to reload your checkpoint saves a lot.
 * Warhammer 40,000; Dawn of War - Dark Crusade

"Prince LaCroix: Therefore, I have decided... [Five minute pause in which he stares around] Nines Rodriguez: This is bullshit!"
 * Happens a lot in Vampire: The Masquerade Bloodlines. In the following lines, Nines was supposed to be cutting off LaCroix.

"Angeal: Zack. ... ... ... Angeal: Embrace your dreams."
 * Possibly only a problem on certain systems.
 * Perhaps LaCroix was merely pausing for effect and Nines was calling him out on it?
 * The Ace Attorney games have this a lot. For one thing, when the judge says that he's declaring a verdict, there's always a pause if someone is going to object at the last minute.
 * Somewhat justified in that they usually do.
 * When there's a text-based interruption, the line being interrupted does not wait for the player to advance the text before the interruption occurs, so this interruption effect works well. The lag with objections is presumably because in Ace Attorney Law School, judges are trained to pause at the last minute, specifically to allow attorneys to object.
 * Final Fantasy after X. There's often pace-breaking pauses in about every dialogue. Here's an example from Crisis Core:

"Marvin: Go! Leon/Claire: But... [long pause while the game loads both the next line and a change in camera position] Marvin: Just go!"
 * He's pausing for emphasis. Evidently a lot of emphasis.
 * Some dialogue files in Crisis Core do take interruptions into account, usually by having the two lines as a single file. Though, you can only find this out via game modding.
 * Averted in Final Fantasy XII in at least one scene where Balthier interrupts Vaan. It's done very naturally.
 * And Final Fantasy XIII keeps the dialogue flowing pretty naturally too.
 * Narmtacular Engrishfest Metal Wolf Chaos, for all its silliness, has a lot of the narration pausing in weird places.
 * In The Elder Scrolls IV: Oblivion, the first (or second) dialog you hear features this trope as one of the Blades tries to explain to his superior what you are doing in your cell.
 * Tales of Innocence suffers terribly from this, between the usual video game woes and a minimum delay between voiced lines of different characters, due to the dialog box design.
 * Tales of Symphonia: Dawn of the New World has a fair bit of this in the first half hour or so.
 * Pretty well averted in Metal Gear Solid 2, where characters interrupt each other pretty much immediately (for a good example of how to do it right, notice how Fatman interrupts Raiden's monologue about how great a person Stillman was, and how Snake cuts Otacon's lecture about what the camera program does). Not so much in Metal Gear Solid 3, though, which was bad because the radio dialogue was a lot more snappy and comical, particularly between Big Boss and Para-Medic.
 * Warcraft III does this a lot, particularly when Arthas is in the cinematic.
 * Every Bioware adventure ever, including Knights of the Old Republic, Mass Effect, and Neverwinter Nights. In one part of the first KoTOR game this is justified; the two lovers trail off as they see angry parents approaching, but before said parents are close enough to shout most effectively. Other cases, less so.
 * In Mass Effect you can choose the next dialogue option while the NPC is talking, which helps keep the conversation flowing. Of course, if you're focusing on the next dialogue option, then you miss what the NPC is actually saying.
 * Mass Effect 2 exacerbates the problem with the addition of dialogue interrupts. The player has to be quick to trigger them, but this (together with the loud 'ding' sound that plays when you do it) means it rarely sounds natural, even for a sudden interjection.
 * One Crowning Moment of Funny uses this to lean on the fourth wall a bit. After your normally fast-talking doctor suddenly bursts into song, even if you select something right away there seems to be a looooong Paused Interrupt... and then he coughs. Shepard is just as speechless as the player.
 * Mass Effect 2 sometimes has the opposite problem on fast PCs, where lines are spoken too quickly on each others' heels to sound natural.
 * Resident Evil 2:

"Sheila: Careful now! That knife's made out of real carbon steel, and it's so sharp it can cut through a... [loading pause] Willy: Yeah, yeah, I know. It's so sharp it can cut through a shoe."
 * Everyone in Dissidia Final Fantasy, but especially Cloud.
 * Happens a lot in the X-series games, mostly with the computer. Changing targets causes it to start over, interrupting whatever it was saying, which leads to much hilarity.
 * All of the dialogue in Eternal Sonata is punctuated with pauses. If even a quarter of a second were shaved off of each pause between lines, cutscenes would be up to 2 minutes shorter.
 * Dragon Age is guilty of this, too, both when the PC is talking to party members and when party members are talking to each other while exploring.
 * The Adventures of Willy Beamish has this exchange, which is much more pronounced in the Sega CD version than the PC one.