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The doctor didn't charge the health insurance in time, am I liable?
If you read all that paperwork they made you fill out at the emergency room, there is probably something in there explicitly stating that you owe any bills you rack up regardless of what happens with the insurance company. They generally have a disclaimer that filing for you with your insurance company is a courtesy service they offer, but they are not obliged to do it.  Ultimately, you are responsible for your bills even if the provider slow-billed you.  Sorry.