This depends on the type of insurance plan you have. Some insurance companies, after paying for your medical bills, seek reimbursement through the “right of subrogation” or through written contractual provisions in your health insurance plan documents that call for reimbursement of any medical bills/expenses paid to you that are incurred due to a tort or third party’s fault. Basically, this means if you get hurt in an accident due to the carelness of another person/company and you decide to pursue a personal injury claim, the insurance company will try to recover the money they paid for your medical bills. However, not every health insurance plan is entitled to this type of reimbursement. For example, entities which are not formed under the ERISA act (traditional health insurers) often cannot recover the medical expenses paid.
Keep in mind that even if your insurance company pursues the cost of your medical bills, there is still a good chance you’ll receive a sizable recovery for pain and suffering, lost wages, and other damages.
To learn more about the right of subrogation and insurance reimbursement/repayment, check out this article.