I think believability, in its most common inference, has to do with what the readers can relate to, whether it is the characters or any other part of the story. Since each reader is different, the way they think a story is believable or not will be different, too.

Coming back to the characters, how a character--be it a dragon or Aunt May--reacts to external events or his own thoughts might make him feel real. That is why in the olden ways of writing, the authors based their characters on real people so they'd know how they would react. Come to think of it, you can base a unicorn's reactions as if he is one of your next-door neighbors.

Then, if the character is a warped or a crazy person, the writer can have the full control of that character's strings. Who knows when, why, or how a crazy person can react, after all!