Yes. Texas recognizes the collateral source rule, which says you can make a claim and get money for your medical bills even if those bills were paid by your personal injury protection (PIP).
The insurance company will probably assert a subrogation right to get reimbursed for a portion of the amount that they paid out. This subrogation concept can be confusing, but don't worry. You can click here to read more on Medical Liens and Subrogation.