Apple can't seem to make up their mind on the issue. I called Apple and they said it was a hardware issue. The phone was considered DOA, and I should swap it at the Apple store for a new one.
I scheduled a genius appointment and they said it was a software issue and he was confident it would be fixed with a restore. Haven't had a chance to do that yet since the person I bought the phone with doesn't have a computer or access to WiFi. I had to purchase an iCloud upgrade, and even in the Apple store it said it was going to take 5 hours to backup the 6GB of stuff on the phone.
I can't say I'm impressed with Apple's handling of this issue.