Ok, found a few consistencies in the log files. About 7 hours into the backup, probably at the end, I get these events:
In the Application log;
Backup Exec Alert: Job Failed
(Server: "SRV-UPDATES") (Job: "Backup 0007") Backup 0007 -- The job failed with the following error: The Backup Exec job engine system service is not responding.
In the System log;
The Backup Exec Job Engine service terminated unexpectedly. It has done this 1 time(s). The following corrective action will be taken in 0 milliseconds: No action.
After a day or two of this, the Backup job engine will be stopped.