There is a statute of limitations ("SOL") for what are, in this case, contractual claims. But its length depends on your state. I am not admitted in Texas, but my research suggests that the SOL in Texas is in fact four years for contractual claims. Considering the size of such a claim, you may want to consult with an attorney on this.
Separate and apart from your strict legal rights, you may also want to think about why you should not pay the bill, although it was rendered very late. You did, after all, receive a valuable and expert service that you presumably needed very badly. Unless you have been billed for interest, you have only benefited from the delay in being billed -- it is as if the doctor lent you the amount of the bill for four years!