Does "predicament" mean something more severe than "problem"?
Not necessarily. A predicament is a difficult, unpleasant, or embarrassing situation. If someone were to say that he has a predicament, it just means that he is in some sort of situation, it doesn't mean that it is more severe than "problem." A: I have a predicament. B: What is it? A: I forgot to bring my lunch. Ex: Her shirt is covered in dirt. That is kind of a predicament. Ex: We are in a predicament. We forgot our IDs before going into the liquor store.