Can I sue for mistakes made on my criminal background investigation?

You could sue the background company for negligence.  Negligence is the failure to exercise due care (that degree of care that in this case a reasonable background company would have exercised under the same or similar circumstances to prevent foreseeable harm).

In order to prove negligence, you will need to prove duty (of due care mentioned above), breach of duty ( failure to exercise due care by the mistakes in the background investigation), actual cause, proximate cause, and damages.  Actual cause means but for the mistakes in the criminal background investigation would you have been denied employment?  If the answer is no, actual cause has been established.  Proximate cause means were there any unforeseeable, intervening events which would relieve the background company of liability?  If the answer is no, proximate cause has been established.  Damages means the amount of compensation you are seeking to recover in your lawsuit for negligence.  Damages would be lost income from not getting the job.  You will need to mitigate (minimize) damages by trying to find another job.  If you find another job that pays less than what you were offered in the job you didn't get due to background company's negligence, your damages would be the difference in wages between the two jobs.  If you don't mitigate damages, your damages will be reduced accordingly.  You will need to file your lawsuit prior to the expiration of the applicable statute of limitations or you will lose your rights forever in the matter.