On 2015 Jan 13 Mail Connection Doctor again reported SMTP password failure.
Clicking on that entry led me to the "Accounts" page of Mail, which showed a 7 character password.
I put in the above 8 character password and AT&T SMPT started working again.

I just had a long conversation with a very knowledgeably person on Chat.
He(?) was as mystified as I. I learned one thing I didn't know: Keychain is in on any password stored in the system.
He was not phased by my distrust of CA's.
Something, probably that, caused screen sharer to fail.
I have "case ID 730133469" in case this should recur. Be sure to notice whether ATT POP is green.
He made notes for future reference, probably keyed to the case ID.
FMI: Mail > Preferences > Accounts > AT&T > Account Information > Outgoing Mail ... > (pull down) > Edit SMTP Sever List... >
or just click window in Connection Doctor

I just now consulted Keychain Access > Keychains > Local Items > Category > Passwords / and noticed that there are two entries under Name for outbound.att.net.
They show the same current password. Under Where the say respectively
smtp://outbound.att.net:465
smtp://outbound.att.net
The later shows that it was altered Jan 11 this year.
This is certainly a source of confusion.
I deleted the 2nd.