Fievel from An American Tail, who suffers disappointment after disappointment as he searches for his family in New York, to the point where he completely gives up trying to search for his family and decides to become a Street Urchin. Cue Gray Rain of Depression as he curls up and cries himself to sleep.
"This is my home now..."
You have to feel sorry for Lucas in The Ant Bully. He's bullied, he has no friends, his parents don't know how to relate to him, his sister shows absolutely no interest in him and he has no other enjoyment but to torture animals all day. And that's before he gets shrunk and kidnapped in the middle of the night by giant ants that push him down various incredibly (comparatively) great heights and then threaten to eat him alive when he is completely helpless. All while he's completely naked. No wonder his eyes start leaking.
The eponymous character of Bambi after losing his mother.
Manolo is mocked by townsfolk for not killing bulls. Looked down on and basically disowned by his own father at one point. Thinks he sees the love of his life die before his eyes. The guy gets put through the wringer.
Maria breaks into sobs when she learns Manolo has died.
Happens to Mater in Cars 2 after he discovers that everyone else views him as a clueless ditz, good only at distracting others while real heroes get things done.
Danny from Cats Dont Dance: He arrives to Hollywood with big dreams and is met with nothing but scorn and cynicism from every other character. He doesn't break, he instead rallies the other animal actors until he rekindles their own dreams.
In Ringing Bell, a lamb named Chirin starts out as cute, friendly, and happy-go-lucky until his mother is killed by a wolf. So he seeks revenge, but realizes that he's too small and weak to do any damage. Thus he convinces the wolf to make Chirin his apprentice and he goes through Training from Hell until he becomes a deformed ruthless killing machine.
The Fox and the Hound: Tod and Copper evolve into a Woobie and a Hero Antagonist through the course of the film. They start out as best childhood friends, and one of them is supposed to kill the other. Tod especially gets it bad; he is abandoned by the only family he knows to live in the forest where he meets some angry critters. He falls in love with a girl fox and makes a fool out of himself in front of her. His best friend blames him and wants to kill him for something that wasn't his fault. The way he got his friend to forgive him? Fight a giant bear and nearly die. And it's implied that they aren't allowed to see each other anymore after that.
Anna and her sister Elsa in Frozen go through this for most of the film: At a young age, Elsa inadvertently almost kills Anna, which forces her to become a recluse who's afraid to show any emotion at all. She won't even let her parents give her a hug anymore, for fear of hurting them. Meanwhile, Anna is left to wonder why her sister and best friend in the world doesn't want anything to do with her anymore. Then their parents die, leaving both of them utterly alone in the world. After the coronation, Elsa's secret is revealed and she is forced to flee the only home she's ever known. Just when she thinks she can live in peace and quiet, she's informed she has just doomed everyone in Arendelle, and she is powerless to stop it even as she starts being hunted down by men who consider her nothing more than a Monster. Anna's one hope to save her after she is left to slowly freeze to death when her sister once again accidentally hits her with her magic turns out to be a heartless manipulator who's only after the throne and leaves her to die. Elsa survives an assassination attempt, only to be imprisoned instead. She finally gives up the will to live when she thinks her sister died because of her, only to be saved by Anna freezing to death.
Megara from Hercules. She pledged service to Hades to save an old boyfriend's life - only to have said boyfriend run after another girl shortly afterwards. Plus, she's implied to have had some run-ins with boys who don't understand the word "no". OUCH.
When Littlefoot in The Land Before Time suffers the death of his mother partially by his own doing. This continually happens when he suffers nothing but disappointment after disappointment.
Simba of The Lion King loses his beloved father Mufasa in a horrifically tragic "accident," which is followed by a heartbreaking Please Wake Up scene when he finds Mufasa's body. Then Scar convinces him that Mufasa's death was his fault and sends him running away from his home and everyone he's known. Not until years later, and a visit from his father's spirit, does he finally stop running from his grief and guilt.
Susan Murphy of Monsters vs. Aliens. Hit by a meteor on her wedding day, she begins glowing green and turns into a monster, sending everyone she knows fleeing in terror. She's captured by the military and locked away permanently. She's with real monsters who don't understand her. When she finally gets a chance to get out (by facing something out of her nightmares), her fiance rejects her and she's abandoned. Then she's kidnapped by an alien who wants the phlebotinum that turned her into Ginormica so he can destroy the world. She finally grows a backbone and takes a level in badass.
Michelle, the badger in Once Upon a Forest. When the toxic gas spreads trough the woods, Michelle ends up passing out while her mom and dad end up at Death's doors. When she comes to, she first feels overjoyed that many of the families are getting back together. But after finding out her parents are dead, her moment ends up getting squashed. While we don't exactly see Michelle sobbing over it (because that would be too much of a Tear Jerker even for this movie), the implications of how Michelle will need to cope with that after the movie ends is tragic enough.
The Powerpuff Girls send themselves on a self-imposed exile after becoming pariahs in the eyes of Townsville and even the Professor. As they commiserate on a lone asteroid, Bubbles is left bawling, Buttercup channels her sadness into rage, and Blossom stares up at the earth with tear-filled eyes, buries her head in her hands and cries to herself.
The normally cheerful and adorable little she-squirrel in The Sword in the Stone is tragically left heartbroken and in tears when Arthur is turned back from a squirrel into his human form.
Wall E. Let's see, the main character gets rejected by his love, multiple times, and ends up risking his life to help her; said girl is called dysfunctional, is classified a rogue robot, and watches the robot she finally loves get squashed; and that's not counting the myriad of possibilities in the repair ward. He's still a pretty cheery guy.
James and Hilda Bloggs of When the Wind Blows are a nice retired British couple who could be your grandparents. They really don't understand the implications of surviving World War III, so we're going to see them die of radiation poisoning, still believing that the Government will collect them.
Vanellope Von Schweetz from Wreck-It Ralph gets this treatment quite a lot. She is persecuted by the other racers of Sugar Rush, her game world, and not allowed to race as she is glitchy and clumsy and players may think the game is broken if she glitches when being played, which may cause the arcade to "unplug" the system (which would mean something of an apocalypse to the game's inhabitants). As an outcast, Vanellope lives alone in a secret, unfinished level; a volcano full of soda and dangerous pop candy. She also has no memory of her past, only that she knows 'racing is in her code', despite never setting foot on a race track. Meeting Ralph helps her cope with this, as they are very much alike in that they exist but don't get enough love from their peers.
Both Vanellope and arguably Ralph himself go through a straighter breaking later, after King Candy convinces Ralph to wreck Vanellope's kart for the good of both herself and the game as a whole - all part of the main villain's plan, as he had messed with the programming with only his own interests in mind. Ralph was only able to un-break Vanellope after he realized that something was amiss in the first place, due to her picture being on the side of her game console.
The character 5 from 9 is a very sweet and trusting character, but that doesn't stop him from getting abused. First he loses his eye. And then his best friend is killed right in front of him. And then, right when it seems like everything is going to be all right, he dies.
Judy Hopps from Zootopia appears to be going through this as seen in one trailer where she starts breaking down in adiwn in her room and starts crying into her arms.