I'm just interested. Since there were many plot twist in books, movies, etc some critics say they're unbelievable. Now I'm asking myself, what makes a plot twist believable/unbelievable?
What makes anything in a story believable or unbelievable? Is it consistent with what we know of human nature, the laws of physics, etc? (If this is a fantasy or science fiction story, is it consistent with the laws of physics et al as this story has set them out to be?) Is it consistent with what has happened in the story up to this point?
You can often make the wildest things believable if you foreshadow them properly.
I heard an interview with a novelist once where he said that he spends a lot of his time when writing putting door in alleys. Of course he said it this way to be cryptic, but then he explained: If you say the hero is being chased by the villain, and he suddenly turns into an alley, and there's a door there that leads him to a place where he's safe, the reader is going to say, "Oh come on! He just HAPPENED to find a door that leads to safety?" But if in a previous chapter you had the hero come to that alley and find that door, and now at a crucial time he runs there and uses the door, it sounds quite plausible.
RE plot twists specifically, I'd add: they must be used sparingly. One plot twist in a story can surprise and entertain the reader. A second plot twist and the reader may say, "wow, you got me again". But beyond that, the story starts to be too unpredictable for the reader to care. I recall a book I read years ago where every chapter there was another plot twist, every chapter we were told that the person or group that we were told were the villains the previous chapter are really good guys, or somebody wasn't who they seemed, etc. After 3 or 4 of these I realized that nothing in the story could be taken at face value. By the time the writer got to the end and had his final plot twist, I was left thinking, "So is this supposed to be the truth now? Or am I supposed to understand that this is a fake too, that the reality is yet something else?" I quit caring long before I reached the end.
@gravity_train has it correct in the comment above:
a plot twist is unbelievable if it comes from absolutely nowhere.
A plot development, twist, and/or character action is unbelievable if it seems arbitrary. If there is no evidence, no foreshadowing, no hints, no clues, absolutely nothing a reader could have picked up, even on a second reading/viewing, to point to the action or event, then it shatters the "willing suspension of disbelief" which a reader must have to embark on a story.
A meta example of this is the satirical film Murder by Death. Parody versions of a group of famous literary detectives are invited to a mansion to solve a murder mystery in exchange for $1 million. At the end of the film, all the detectives assemble and one after another accuse one character of committing the murder(s). But the character stands up and snipes at all of them:
You've all been so clever for so long you've forgotten to be humble. You've tricked and fooled your readers for years. You've tortured us with surprise endings that made no sense. You've introduced characters at the end that weren't in the book before! You've withheld clues and information that made it impossible for us to guess who did it.
Additionally, when a character acts "out of character" by doing something without any kind of motive, or acting against what's been established for the character previously, without explanation, the audience loses belief. "Without explanation" is the important part — over the years of Star Trek, Vulcans (who are presented as logical, stoical, and generally imperturbable) have laughed, wept, danced, screamed, and raged, but all believably, because the script made a point of explaining why the Vulcan in question was behaving that way.