Depends on where you want your support to come from.
If you've damaged your phone and make a claim on AT&T insurance, you're likely getting an iPhone that was refurbished by the insurance company. It's likely a repaired phone, and cellular insurance companies use third parties to make the repairs. Some might be Apple-certified. Some definitely aren't.
The problem arises if you end up having an issue with your "new" phone that should be under warranty. I've seen more than a few cases where people have brought their carrier-insurance-refurbed phone to an Apple store because something was wrong with it, and the store wouldn't touch it because non-Apple parts were used to make the repairs. At that point, you no longer have a warranty, and pretty much have to try your luck with the carrier to get your phone fixed again.