Is it possible to get a misdemeanor petty theft off my record?

I took the rap 16 years ago for a crime I didn't commit. My cousin was already in trouble with the law and on probation she would of had done three years if she was convicted. I was never in trouble with the law and I was young and dumb at the time. I was 19 years old when this happened. I never thought about how this would ruin my chances in getting a job in my future. I recently graduated in the medical field. I found a job and I did not think I had this on my record. I got fingerprints done waited for my background check and come to find out after 16 years, a misdemeanor pops up and how embarrassed was I and very sad. I never have been in trouble with the law since that time. Please tell me there's hope.