No, there is not statute of limitations on owing a bill. If you do owe it they can bill you at any time. They do have some regulations as to how they have to handle it though. Such as when they can turn it in to a collection agency, when last contact with you about it is, etc....if they contact you now and you don't pay they can still keep it for 7 years then file it with a collection agency.
That's what my boss found out about some unpaid bills she sent to a collection agency. They told her she could do all kinds of things.
If you owe someone money it never goes away, it is still owed. They can take you to court, file a lien, get garnishment, etc....for years.
One of the odd things that was posted by a FB lady that owes money to my boss was that she was going to just wait it out...until the bill was not able to be collected on. Like a statute of limitations on crime. A couple of attorney's corrected that person by mentioning this. IF IF IF a bill could be "waited out" and the person gets out of paying it wouldn't everyone who ran up huge debt just do that? Instead of having to file bankruptcy? OR go to court and have garnishments?
Call them and talk to the office manager/billing office. They should be able to give you all the information you need.
I imagine there was some sort of error about the whole thing. They need to be able to produce an itemized bill, showing who they billed it to, what the insurance company said about why they were being refused payment, and if anyone even filed or received payment and didn't write it down.
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Response to your ETA
Call the doc's office that did the procedure. This does sound like a collection agency for doc's.
That doc's office should have every bit of information you need. They may say they can't talk to you about the bill if it's turned over to collections, then you'll know for sure.
It could be a scam too. These days one never knows. I'd call the doc's office to make sure it was even legit.