Hollywood Encryption

Modern cryptography can be what protects the citizens against The Man. It can be the difference between losing a war or winning it, the difference between making millions of dollars off a new invention or letting the competition find out, overtake you and win all the money you could have made. It can be what keeps your Porn Stash safe from Mom's prying eyes. It's the only barrier that separates data crackers and your bank account's details, and it can be the base of communication within an Ancient Conspiracy or a shadowy criminal organization.

Unfortunately for those who studied literature hoping to never meet a single number ever again, cryptography is also one of the hardest sciences in the world, drawing from fields as abstract and esoteric as number theory, mathematical logic, information theory and data structures. Needless to say, this means research in the subject needs to be very accurate. So when your executives are demanding that you deliver a manuscript in record time, the only thing you can do is depict it as some kind of character shifting or describe it through some kind of Techno Babble.

Can be an acceptable break from reality, because there would be no plot at all if an encryption really was unbreakable or took years to be broken.

Anime and Manga

 * Done in Ghost in The Shell, with the Hand Wave that the Major and the more techie members of Section 9 are just that good. However, sometimes they run across security that actually counterattacks their hacking, which might be something of an aversion (encryption that can scramble your brain? AWESOME).

Film

 * Averted thoroughly in Sneakers. While the MacGuffin is entirely fictional, if something did what it could do, it would be able to slice through virtually any modern encryption.

Literature

 * In Digital Fortress the NSA has a computer which is powerful enough to brute force (i.e. keep trying different passkeys until it gets the right answer) encryption. The plot is based around a new encryption algorithm which is resistant to brute force methods. This is a clear case of Did Not Do Research, since a brute force search for a solution would try every possible key until the right one was found.
 * There does, however, exist an encryption method, the one-time pad, that is immune to brute force, and unbreakable if carried out correctly (doing so, however, is often logistically prohibitive). It's immune to brute force because with different keys you can get every possible message that has the same length as the one being sent.
 * Actually most modern digital encryption is resistant to brute forcing, in that it's theoretically possible to do it, but would likely take an impossibly long time, possibly billions of years, assuming you dedicate all of the worlds computing power to brute forcing that one key.
 * Averting it is pretty much the entire point of Neal Stephenson's Cryptonomicon, where Lawrence Waterhouse shows us how the German Enigma machines worked and how Bletchley Park worked day after day cracking their codes, whereas Randy Waterhouse shows us in the first few pages how proper encryption is done, all of that while rattling scientifically accurate lectures about information theory. Need anything more? Well, Neal Stephenson asked Bruce Schneier to create a 100% functional encryption algorithm, called Solitaire, which can be implemented on the field with playing cards, and the books include a working Perl script that implements the algorithm.
 * In Desmond Bagley's The Tightrope Men, the hero is captured and being questioned by enemy agents, who want to know about the high-tech whatsit he's believed to be working on. However, he's only impersonating the scientist they think he is, and he didn't get a proper briefing, so he's at a loss. Then, abruptly, he comes out with a spray of technobabble about a computer which can brute-force encryption like the one in the Dan Brown example above. This scares the life out of him, because he doesn't even understand what he's saying, but it does make sense to people who know computers, so he wonders where the hell this sudden burst of information came from. Incidentally, the scientist he's posing as was actually looking into a possible design for an X-ray laser, nothing to do with computers at all.

Live Action TV

 * Stargate Atlantis: Multiple times. Janus' lab pops up. He is meant to have his research encrypted with highly advanced encryption. It doesn't last long against the mind of The Smart Guy.
 * The Sarah Connor Chronicles: Of course can John break the encryption of Sarkissian's hdd.
 * Willow of Buffy the Vampire Slayer fame can decrypt anything, seriously, though admittedly the Initiative's most secretest files took her a few days...
 * Actually, she didn't manage to decrypt them; they eventually decrypted themselves, which was a less-than-subtle clue that the season's Big Bad had slipped those files to the good guys intentionally.

Video Games

 * Averted in Iji. The eponymous Nanotech cyborg can hack her way through a lot of low-security doors, but some doors are just too securely encrypted for her to ever hope to open.