The statute of limitations for medical bills is the same as for any other contract - 4 years.
However, the statute defines whether the creditor can sue to collect, not whether they can report the debt as unpaid. Despite many companys that claim to be able to remove valid debts from a person's credit report, there is no legitimate way to do that.
Under the Texas Civil Practice & Remedies Code, the statute of limitations to be sued for most debts in Texas is 4 years. From the facts included in your question, all of these invoices are too old to be sued upon.
But according to the federal Fair Credit Reporting Act (FCRA), unpaid accounts can be reported on your credit reports for 7 years. Therefore these will not drop off of your credit files until at least 7 years have gone by. If they have not dropped off by then, you should dispute the accounts with each of the three major credit bureaus, and indicate to the credit bureaus that you believe the accounts to be "obsolete," i.e. too old to continue to be reported.