I have BPD and I'm in school right now working on becoming a psychologist. My therapist says I'm going to be a fantastic one (I wish I could believe him) so yes, it's definitely possible.

It's important to know your own limitations and be actively working on your own issues, doing everything you can to guard your own mental health and stay on top of your illness(es).

They say "to BE a good psychologist, you have to HAVE a good psychologist". I firmly believe that. Most therapists see another therapist, and some therapists specialize in only working with other therapists.