Try stopping the Inbound and Background services completely and start them back up fresh. I have seen cases where the service actually will fail and windows won't log it in the event viewer log.
I restarted both previously and thought that might have helped, but the issue returned. This time I did a stop/start on both and will see if that works better. So far so good as the issue had just happened again as I wrote my question. If the problem is in fact with the mail services, at least I know that should be resolved when we migrate off our current app server.
We have had similar issues, on the past couple of occasions emails in TNEF format completely stopped the inbound service, workaround being to identify and move that e-mail. More info here: https://community.landesk.com/support/docs/DOC-10731#comment-23032
I have an open support call regards this issue as we have other e-mails in the same format (tnef) which are processed.
We have implemented the ticker on Service Desk Clients, it display a the count of number of tickets created via e-mail within the last hour, if the query returns 0 then is displays in red and we can quickly look to resolve the issue. In the past we didn't find out until half a day leading to a massive backlog. The resolution as mentioned before could be moving the tnef formatted e-mail, or as Jamie has instructed.
Thanks, Afzal. I reviewed the linked doc and I've not seen that error in our event logs on the server. In our case Service Desk will eventually start creating tickets again from emails after a rather long delay. But kicking it by stop/starting the mail services seems to work.
Since we're still on 7.7.3 I'm hoping moving up to the latest versions will help alleviate this problem altogether.