If a Bat Kohen marries an am haaretz the Gemara (Pesachim 49) says problems will come. Some say that if you marry a bat kohen you have to be extra careful in how you treat her. In fact even scholars may avoid marrying a bat kohen. I read that he Abuchatseria (sp?) family – that of the Babi Sali – avoided marrying b’not kohanim.