Can I be billed for a doctor’s visit from over 5 years ago?

It *may* be too  late for them to enforce the debt against you. The statute of  limitations for contracts--and this is essentially a contract action; it's based on the doctor's terms of service, any agreements signed when you became a patient, etc.--is 5 years in your state. The cause of action acrues when the debt is not paid on time, which given usual payment terms, was probably 30 days after the amount was due. If the action accrued more than 5 years ago, barring some special circumstances, it may be too late for them to sue you if you refuse to pay; if they can't sue you, you could ignore the debt.

Even if they can sue--that is, the statute did not quite run out for them by the time they bring a lawsuit (if they bring a lawsuit)--there are other doctrines, such as "laches," that may let you contest a too-old lawsuit ("laches" is the doctrine that if a would-be plaintiff waits unreasonably long to bring an action, which prejudices a defendant since it's harder to find evidence, documentation, etc. to contest overly old charges, the courts may consider the plaintiff's actions inequitable and dismiss the case).

Bearing the above in mind, for small bills, it's probably easiest to pay to avoid the headache of having to defend yourself. For a bill larger than your willing to pay for convenience's sake, you could refuse to pay; only if they sue you, would you have to defend; and if they do sue, you can then see if either the statute of limitations or laches helps you; and if they don't, then you might want at that point to agree to pay to settle.