Do I still have a claim if my medical bills were paid by insurance?

Yes. If you file a lawsuit, the total cost of  medical bills will be calculated as part of the damages. This sum includes whatever your insurer paid to cover your expenses. If you receive a settlement or a judgment in your favor, however, you may need to reimburse your insurer for what they paid on your behalf.  This is a contractual right, called “subrogation,” that many insurance companies put in their contract with you.  Depending on the type of insurance, some jurisdictions limit the right of insurance companies to bring these subrogation claims.