There's usually a process to handle negative information that comes up.
Court and police record keeping is poor and identities aren't always verified --- if someone is arested and doesn't have ID, but it's not worth charging them with anything, you let them go in the morning, and file it under whatever info they gave.
So any big company has gotten false positives and knows how to deal with it. Depending on the details, they may delay or pause employment while the background check company does the legwork to get more details on the negative information to see if they can determine if it's actually you.
Are you lying about having a degree?
The main purpose of a background check to confirm what you told the employer. If you were hired on the strength of your work history and were honest about your education, felony status, etc. you are fine, otherwise you are busted.