Once an admin has an account setup, it could stop working in the future for a variety of reasons: an SSL certificate could expire, DNS lookups could fail, the mailbox may be full / have a reset password / etc.
Currently there's nothing to notify an admin when that happens, and they may not realize for awhile, which could cause problems for them.
What would be a good way to notify the admin? E-mail? An admin notice banner in wp-admin?
We should probably wait until it's been unavailable for ~30 minutes before issuing a warning, to avoid nagging over temporary failures.