I spoke to Apple support this morning. The rep's supervisor said to try setting up the AT&T account as a Yahoo account. This didn't work because in the verification process, it asks to log into the Yahoo account, which apparently is not the same as the AT&T Yahoo account. The user has only an AT&T account and no Yahoo account.
Trying to set up the account under Other results in an automatic default to an IMAP account instead of giving me the choice of POP or IMAP. The verification process fails because it can't find the IMAP server.
I tried setting up this account on my own iPad 3 with the same results. When I got my new iPad 3 a few days ago, I set up everything from a backup of an iPad 1, so I didn't have to add any AT&T email accounts, and my AT&T email works fine. The problem seems to be when you have to set up a new AT&T email account under iOS 5.1, possibly only in connection with a new iPad 3.
The owner of the troubled iPad 3 has an older iPhone under which the email account works, so I am going to try to use an iPhone backup (which doesn't currently exist) to transfer the email account and other settings to the new iPad 3.