Well, I hope this will help other people - apparently the problem I had had nothing to do with Backup Exec. I kept narrowing down my selection of directories until I was down to 3 directories (off of the root directory.) So I figured I would just zip up those directories and backup the drive with the zip files. Well, when I tried to zip up the first one, my server rebooted. I was even running the WinZip program from a workstation, not even on the server itself.
The directory that was giving me trouble had a whole lot of zero-length and small sized files. I had never had a problem copying the files from the drive (running an XCOPY disk-to-disk backup), but something about the way that WinZIP and BackupExec access the directory caused a reboot.
Fortunately, I have that data elsewhere, so I was able to delete that directory (although I did notice some minor weirdness deleting the files) without crashing the server. Ever since, my backups run through fine with no more problems.
I previously was seeing the same thing that pete405 reported above - all that shows up in the event log is a message saying the server had unexpectedly rebooted. Once while I happened to be in front of the server at the right moment, I did notice a BSOD screen saying "Page Fault in non-paged area" and it gave a filename of NAVAP.SYS. That's all I was able to write down before the screen went away and it rebooted. Anyway, I hope that will help others who are having this rebooting problem. Perhaps they will also find that BE is not the actual problem, it just happens to be an application that triggers the problem.