Is it legal to "double dip" medical payments when settling auto accident medical bills?

Question Details:

I was rear-ended and the party that hit me was 100% at fault. I have heard that it is legal to pay my medical bills through my own insurance with co-pays and then at the settlement submit all my bills, in full, and they will be paid again. Something about that sounds fishy. Can I double dip here if it’s within the law?

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