+1 to AL's response ...
Another 'stab in the dark' ... and a suggestion for some sluething of the backup file that is created.
If you have command line access, there is a script for backing up a single course which will take parameters in the Moodle Admin UI for what to include in a backup. Location: /moodlecode/admin/cli/ it's called 'backup.php'. Run the script by giving it the course ID and I'd also have the script save the backup to a directory outside moodledata/filedir/ (hard to find in there for sluething purposes). Just run the script the first time with no parameters and a help screen will appear describing the parameters it will accept.
Once the backup is made, change into the designated directory and un-compress the backup file. Those .mbz files are really .tar.gz's now so one will have to use a compatible command: tar zxvf [nameofbackupfile].mbz.
If successful in un-compressing you should see the .xml file shared, but the data for student responses, etc. might be in the activities directory. Forums and other activities are stored there. The directory there might be named 'socialwki' and contained there in might be by student ID or context ID directories and those might contain the actual text of their responses/additions, etc. (that's the way the forums work so am making assumption the wiki would do the same ... hmmmm 'assume'!)
There might also be a .log file at the root of the un-compressed backup which could be revealing .... have seen references to 'unsupported addons' and rather than be a show stopper, the backup routine skips those but reports them in the .log file.
One could actually see the size of the big chunks of data Moodle is having to work with IF the system can deal with creating the backup.
Check server logs ... apache error or even mysql error logs for clues.
Also, you've not described anything of the OS software upon which those Moodles run. Yes, you might have the minimally required PHP, MySQL, or whatever, but if the box is a 32bit machine, strange things happen which may/may not throw errors you can see.
Might turn on debugging and step through a baciup of one of those courses using the wiki. See what it says, if anything.
'spirit of sharing', Ken