Hi Barry,
You can check the rules-process logs based on the activity id and see what happened to it when it first entered the system
When this happens again you can also issue the following command to see the status and substatus of the activity
select * from egpl_casemgmt_activity where activity_id='id of the activity'
let me know if this helps,
Christos
Edit:
Actually I checked the image you have uploaded.
An activity will be delivered to the exception queue, with the audit trail showing that it is "Activity hard undeliverable" because a phrase in the subject or email address has an entry in the Delivery Exceptions configuration Pane, with "Permanent" in the Failure Column.
Login to pa-->Admistrator console-->Department-->goto the respective department-->Email-->Delivery Exception and check all the "permanent" failure type
exceptions.
Does this make sense ?