Avon, during the show's fourth and final series. The final straw comes with Blake's (apparent) betrayal. Considering what he's been through, the only wonder is it didn't happen sooner.
Also Travis, who starts off as cruel and obsessive, but rational, and ends up as a homicidal maniac who sells humanity out to alien invaders just to get a chance to kill Blake.
Breaking Bad: Walter White display this at the end of "Crawl Space" laughing maniacally after realising that Gus Fring is going to kill his family and Skyler gave his money to Ted Baneke. Many feel that this is the point where Walter White completely died and only Heisenberg remained.
Buffy the Vampire Slayer: Several examples. Faith, portrayed as not the most stable of individuals to begin with, begins by trying to rape Xander to death and it goes downhill from there. After Tara's death Willow thinks nothing of killing her friends if they get in her way of vengeance, before deciding that destroying the world would be a Mercy Kill. Even Buffy herself is prone to this: the stress of each season finale seems to bring out the worst in her (such as trying to drive her friends away, fleeing from Sunnydale, thinking she can't let anyone close out of fear of getting hurt, then It Gets Worse, oh boy does it ever get worse.
Dexter wrestles with Sanity Slippage during the second season of the series.
Mrs Tishell on Doc Martin. Her crush on Martin went from "she's an odd one", to "what're those pills she's taking?", to Stalker Shrine, and finally a full blown psychotic break, which was a bad interaction between two drugs she was taking.
The Master has always been rather unstable, try as he might to hide it under his veneer of a Magnificent Bastard. However, this takes a turn for the worse in NuWho, when he complains of a persistent drumming in his head, getting even worse after returning from the dead in the Specials Season. The main question is whether or not the drumming is real, making the Master even crazier (It is, turning out to be part of a Gambit Roulette by the Lord President of Gallifrey to free Gallifrey from the time-locked Time War). With this revealed, fans speculated on how bad the drumming was during the Master's previous incarnations.
Firefly: One of the tie-in bits of promotional material for the movie Serenity is a short series of films called the "R. Tam Sessions," which depict River's traumatic time at the Academy. It starts off with River being a happy, eager girl who really wants to learn and push herself to her limits, but as the series progresses, she is shown slipping into madness due to the Academy's experiments. The series ends with River killing the man who has been interviewing her by shoving a pen through his throat (who, for those of you playing at home, was played by Joss Whedon).
Specifically, Will has extreme empathy that he uses to solve murders, and encephalitis. Will's psychiatrist is a serial killer. Will's psychiatrist knows he has encephalitis, but as it worsens convinces Will that his hallucinations, sleepwalking, lost time and seizures have no physical cause. Will's psychiatrist encourages him to think that killing a different serial killer in the first episode felt good, manipulates him into shooting another serial killer while hallucinating the previous one, and frames him for multiple murders. Will actually copes remarkably well, given the circumstances.
Heroes: During Volume 4, Sylar's already-unstable psyche takes a dramatic turn for the worse when he acquires the ability to shapeshift and starts to lose his sense of identity after using it repeatedly.
Mash: Although it's never shown, during the finale, after witnessing a traumatic event, Hawkeye begins to not-so-subtly show signs of a breakdown. Like being convinced one of the anesthesiologists was attempting to suffocate his patient with the gas mask, or crashing a jeep through the mess tent. Little indications of a stressful work environment.
Merlin: Morgana starts out pretty sane, but after she goes evil she starts going off the edge. She is encouraged by her sister to isolate herself from everyone nonmagical, and when her sister dies becomes obsessive about gaining her revenge. During the Time Skip between seasons four and five, she was locked at the bottom of a well and didn't see the light for two years, which has damaged her beyond all repair. Even Mordred goes from idolizing her to being frightened by how unhinged she is and stabs her in the back when she tries to kill Arthur.
Simon Bellamy from this British sci-fi series is a painfully shy and intense introvert (and convicted arsonist) who has been bullied and ignored all his life; it's hinted from the onset that he's teetering on the verge of mental breakdown. As the series continues he accidentally becomes party to murder, is magically imbued with the power of invisibility (which, awesome as it sounds, greatly heightens his growing sense of alienation and his unhealthy tendencies towards voyeurism) and inadvertently causes the death of Sally - a woman he was starting to fall in love with. During the final episode of Season 1, the full extent of his sanity slippage is magnificently depicted as he casually munches on some left-over pizza while gazing serenely at Sally's corpse, which he has propped up in a large freezer. He got better though.
Series one also shows Sally the probation worker go through a (arguably milder) version of this. Tony, the previous probation worker was her fiance and she spends the whole season trying to prove that the gang is behind his mysterious disappearance. (and she's right about it), though she spends most her time stalking the misfits and by the end became completely obsessed by this.
Since this is a post-apocalyptic setting, it's a given that a lot of characters have experienced this.
It's suggested by several characters involved in the militia (like Jeremy Baker in episode 3) that Monroe's been suffering from this somewhat ever since Miles left him. episode 10 is when this trope becomes obvious for Bass Monroe. In episode 15, Emma Bennett, Monroe's high-school girlfriend, dies before she could tell him where to find their son, and that had to have added to his insanity. The first season finale reveals that it wasn't just Miles leaving him. It was also the fact that Miles tried to kill him while he was sleeping and leaving without an explanation. The fact that Monroe has a borderline erotic fixation on Miles did not help matters.
Ray Kinsey in episode 4 fell victim to this as a result of watching his daughter die.
Drexel in episode 6 has this going on, but he is a sociopathic drug lord who has likely become addicted to his own heroin.
In episode 12, Randall may have undergone one triggered by the death of his son, resulting in him co-opting the Mathesons' research to produce a superweapon—initially to end the war in Afghanistan but ultimately to destroy the world order and start anew.
Rachel Matheson has been undergoing this since the blackout, as revealed in a flashback in episode 19. It only got worse after the death of her son Danny in episode 11.
Smallville: Major Zod of the 9th Season started out bad and only got worse from there.
Antagonist Dukat. He starts off as a scheming, megalomaniac, yet charming villain, gets gradually worse, especially in season six, until it all finally boils down to this: "I should've killed every last one of them! I should've turned their planet into a graveyard the likes of which the galaxy had never seen! I should've killed them all."
Faked—probably—by Sisko in For the Uniform. He even slips into the Joran voice.
Has some contestants who seem to have gone a little nuts during their time in the wilderness:
Survivor: Amazon had Matthew who freaked out his teammates by beginning to very meticulously sharpen the machete every day while staring creepily out into space, and Butch who suddenly gained a firewood fetish and forced his teammates to stockpile all the wood they could find which ended up burning their camp down when said massive woodpile caught fire.
Butch and the firewood. Even Matt thought that was a little messed up.
Panama had Shane who made the genius decision to quit a 3-pack smoking habit literally hours before getting on Survivor. Cue him yelling at his tribemates every single opportunity he got, treating a hunk of wood like a Blackberry, and looking more and more like a crazed animal the longer he stayed out there.
Heroes vs. Villains saw Russell get hit hard with this trope. Parvati thought it would be just peachy to keep a hidden immunity idol from him just to see him squirm. Once she handed them to Jerri and Sandra to nullify all of the Heroes' votes for the former, the Disaster Dominoes began. For both sides.
Only Sane Man Alex fell into this whenever he really got over the edge. In an early episode, Alex was given some pills he thought that were aspirin, but these were actually "uppers". Their effects are practically immediate, with Alex talking very fast and singing "Bye-Bye, Love" after waving goodbye to the rest of the gang when he goes to see Mr. McKenzie. Later he ripped Louie's cage with his bare hands and threatened Louie because of his refusal to listen to him about re-hiring Bobby.