Resting on real facts or sound thinking, so it is reasonable to hold. It is used about the things inside your head — a fear, a suspicion, a belief, a worry, a complaint, a hope. Saying someone's fear is well-founded is a way of saying 'you are right to feel that; there is something real behind it.' The picture inside the word is a building: a foundation is the base under a house, and a well-founded belief has solid ground under it, while a groundless one is floating on nothing. English keeps that picture alive in the opposite words too — unfounded, groundless, baseless. The word is formal, and it appears most in news, law and careful writing; in speech people are more likely to say 'she had good reason to worry'. One useful thing for students: it does not mean the belief is proved true. It means there was enough evidence to make holding it sensible. A well-founded fear can still turn out to be about nothing.