A Linux and programming background, although very beneficial, is not really useful in this instance... "tinkering" at the system level on a phone system is asking for problems. These systems are highly specialized and streamlined, you should not venture into them yourself and you should contact your authorized vendor for assistance. Even certified technicians do not venture deep into the system, too many components interact with each other...
Basically, I am saying reboot... if that doesn't fix it contact your service provider to get Mitel involved to see what the issue is. Oh, and if you have to reboot more than a few times a year, contact your vendor too, because something deeper is wrong.
Do you know what mailbox is causing this issue? If so, remove it and add it back in again, might clear it. Otherwise I would get your vendor involved.