That a real pain. Even the most saintly people I know (no parking tickets, no overdue library books etc.) sometimes find out that the FBI or Homeland Security or the local PD have something on them. It's a huge inconvenience to get it cleared up, but it's doable. You may need a lawyer. After that keep ALL your records at hand. Examples of stuff that came up on some people's background checks:
Noise complaints (turned out the neighbors were lunatics)
Dismissed charged that were supposed to be expunged from record.
Person flying on a business trip with a business associate who had the exact same name as a terrorist (but was not the terrorist)
Mistaken identidy, case dismissed
Car in hit and run incorrectly identified as belonging to applicant
And on and on. Take the time and expense to get a lawyer to help you. This may follow you for the rest of your life.
But it is indeed fixable. If you can find a lawyer who is connected politically, that helps.