For one of my configured email accounts in GMail I send mail using my university's SMTP sever. This works fine most of the time. When my university password expired GMail was no longer able to send mail via SMTP, but it provided no feedback indicating problems. I wrote and 'sent' messages for a week with GMail before noticing (due to lack of receipt) that they weren't actually going out. Is there any built-in error log or other mechanism in GMail that I can monitor to be alerted of sending errors or to help troubleshoot problems?
I think something is off with your university's mail system. I've tried sending mail using a custom From address and an incorrect password and it's been my experience that Google's mail system tells me that the remote server rejected my credentials via an error message delivered to my Gmail inbox. For example:
Delivery to the following recipient failed permanently:
Technical details of permanent failure: Google tried to deliver your message, but it was rejected by the recipient domain. We recommend contacting the other email provider for further information about the cause of this error. The error that the other server returned was: 535 SMTP AUTH failed with the remote server. (state 8).
Two things to try out: 1) Run a quick test in Thunderbird Portable to see what sort of error message might come back from your school's server if you send it incorrect credentials. 2) Contact the IT department and ask them if you should be receiving errors from their system.
It's also possible that you have a filter that archives or deletes messages matching some criterion that Google's errors happen to share. Give your filters a once-over just in case.