Yes, they can: if they were not paid by insurance (e.g. insurance took the money back), you are responsible to pay for your treatment. Their claim against you is considered one arising in or coming out of contract: out of the agreement that they would be paid in exchange for treating you. The statute of limitations, or time within which to sue, in your state for a claim arising in contract is at least 5 years, which means they have at least 5 years from the appointment to take legal action to recover the money. Therefore, they are well within time to "come after you."