Hello Beco,
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Did you read Jessie release notes before upgrading and upgrade procedures?
https://www.debian.org/releases/stable/amd64/release-notes/index.en.html
Since you teach students, it would be good to teach them "best practices" by running well organised and prepared server...
I'd re-think the partition layout:
https://www.debian.org/releases/lenny/ia64/apcs03.html.en, second paragraph:
For multi-user systems or systems with lots of disk space, it's best to put /usr, /var, /tmp, and /home each on their own partitions separate from the / partition.
Are there any backups / RAID for users' home directories?
Please don't consider it as offensive, I'd like just to let you know that the problems you've encountered could have been much worse
Good luck ;-)
Cheers
Marek