Can send but not receive email on an AT&T and an SNET account.

This is an old issue that has never been resolved, as far as I can tell from posts going back at least 5 years.  These old accounts can send email but all emails sent to them bounce at the server with an error:

The response from the remote server was:

550 5.2.1 <[redacted]@att.net>... blank mailhost - invalid address, relay=[209.85.222.175]

To further clarify, I can log in fine, and the accounts send emails normally.  This is not a client problem, since it occurs with webmail as well as with email applications.

I have one SNET account that still works, probably because I never paused using it on a regular basis, but several others are in the state described here.

Any ideas on how to resurrect these?