Understand your concern, however, a little more clarification or info might be needed.
Are we talking a course on a school server (site) or an entire moodle implementation ... a site? Am going to assume a course on a school moodle site.
For others ... version 2.6.? of Moodle on Windows (version?) with PHP 5.4 (if I've found the right site)
The error link does suggest that administrator turn on debugging. Has technology coordinator done that? If he/she has, what did (if anything) did debugging report when attempting to backup your course? Does the error happen after the backup has started? There are a few screens one has to navigate through to backup a course. Do you get all the way to the end of selecting options and the button that says 'perform backup'?
Suggestions for technology coordinator/server administrator:
check messages log (if Windows keeps such a log) or the mysql logs for clues.
using some DB tool, like phpmyadmin, check into the size of tables related to backups. Could be the
mdl_backup_logs table has grown very large. Too large, in fact.
check moodledata/temp/backup
That's the folder used by the backup routine. In there, one might find a larger then 0 byte log file that is text. It might contain clues. There could also be directories with long named ... bunch of letters numbers ... that contain the pieces to your course backup. If the backup is large, it actually might be there ... one will see a .mbz file that server couldn't copy to the area where Moodle is configured to save backups.
'spirit of sharing', Ken