Can a hospital send you a bill 3 years after service, if you thought that your insurer had already paid it?

Yes, the hospital can. The statute of limitations, or time to sue, for an action based on a contract (such as the agreement or paperwork you did to get medical treatment) is 6 years in your state. That means that the hospital can bill you  for the services--and, if you don't pay, sue you--up to 6 years after the service was provided. It does not matter whether you thought the insurer paid, or that you didn't owe the money for some other reason: if the bill has not yet been paid, they can send it to you.