I don't really like this name. There's always the Romeo and Juliet factor, plus it just doesn't sound all that good. It would be nice on a cat, though. Tybalt would be a nice companion for Miss Kitty Fantastica.Plus, it also makes me think of Tybalt from the webcomic Boy Meets Boy. If BMB ever really had an antagonist, it was Tybalt (or Collin, but that's beside the point).
This name makes me think of someone who is fiery and angry, probably due to the character in Romeo and Juliet. Tybalt had it comming to him when Romeo killed him. He was a jerk. I wouldn't use this name.