It's very, very possible.
My mom I'm sure had some type of anxiety disorder. She had this nervousness that would cause her to like, shake her legs. Like.... she would be sitting and then knock her knees together in continuous motion, like a nervous tick or something. She's passed away so I can't discuss it with her, and to me it was normal because she was my mom. But, now that I'm grown that's one of the things I would like to ask her about. Other people thought it was "weird." My father even once said that's probably why she didn't date much before him.
My father.... he's like one huge bowl of mental illness.... I know he's had sever depression probably since my mom died. He's on depression meds now. But, I would not be shocked to find out it was bipolar. He had some major anger when I was a kid. Drop of the hat type anger. He also has super high anxiety, like to the point that I think a couple of his heart attacks are a direct result (because of his other ill health, like it pushed him over the edge.) I think that leads to paranoia, which I know he has.
It's his first cousin who has schizophrenia. I don't know what type of schizophrenia but I know it caused him to be violent, so he's in that small percentage. On meds you can't tell, just seems like a quiet person, but since his dad died a couple years ago he's gone down hill. I am sad for him, because he's super nice but once his mom dies, I don't know if his siblings will take him in.

And my ucle (father's brother,) had some sort of sever mental health issues. Like, to the point that he not only put himself into isolation but also his entire family. Since he passed away in June, my aunt has become quite the social butterfly. My two boy cousins on that side I don't know much about them in that area, but my aunt told me my girl cousin has the same mental illness as my uncle, but she didn't say what it is. But she has very sever anxiety.
My mom's brother has sever paranoia issues and I would not be surprised if he has bipolar or schizophrenia or schizoid or something. He is very isolationsit, even against the family at times. He's had delusions in the past, never married, and no kids, either.
So yeah, mental health issues do run in families.