To "bail" on someone can mean to leave but in a specific type of situation. For example - if someone leaves their friend to cope with a situation by themselves e.g some people on a night out are joined by some annoying guests, one of the people decides to leave, leaving his other friends to cope with the annoying guests. That would be considered bailing. So, generally it means to leave early in a way that forces your responsibilities onto other people.

to bail = to quit