It's very possible. Every year those investigating such things find more and more things that are linked to hereditary causes. Mental illness changes the way your brain works, depending on the illness, and it's hard to think that such changes don't also do something to our DNA. Which, naturally, would make any offspring either have the disease (such as the case with things like dwarfism) or be more susceptible to it (high blood pressure, high cholesterol, ect).
So far I don't believe there are any 100% confirmed mental illnesses that are hereditary, but it also wouldn't surprise me as we take steps forward and understand more and more of how things are passed down from one generation to the next.