While I agree that "success" is an ambiguous term, mental illness does not go hand-in-hand with poverty.
Ironically, my job and the agency I work for is dedicated to assisting individuals with disabilities (including mental health) prepare for, enter in to, and advance in employment. If you live in the US, it is called vocational rehabilitation and can help you!
I started out as an assistant and have since put myself through school while working full time to become a counselor. It is INCREDIBLY demanding and there are days I feel overwhelmed and have to just take a day or two and regroup. During my work here, I still struggle with my depression and anxiety. But, I told myself that there is nothing that is going to keep me from helping others and accomplishing my dream.
Ironically, my professional life is much more stable than my personal life. I really throw my entire soul in to work and come home exhausted each day, but I am making a difference in other peoples lives and it keeps me going. I've been divorced and re-married and my debt goes up and down like a roller coaster, but with each promotion I've gotten and each step I take at work, the less and less wild this coaster is.
So, don't let it stop you! And, know that if you are in the USA, there is a state funded agency (not to mention private as well) whose sole purpose is to help individuals with mental health diagnoses get GOOD, STABLE jobs.