I think the error message means exactly what it says: the JavaScript in the student's browser tired to contact the Moodle server to save the current responses, and it failed to get through. Therefore, there is probably a network problem, or a problem with the Moodle server.
Just because the IT people did not see any other symptoms of a problem does not mean that there wasn't one.
To know what actually happened, you really need to find some logs. E.g. do these autosave requests show up in the Apache logs on the Moodle server? If not, then definitely a network problem. (Well, what you are likely to see is a whole lot of autosave requests up to a certain time, then they stop.)
Or, the requests might be in the logs, but might suddenly change from 200 OK responses to a different status code. In that case, are there any errors in the PHP log?
Also, do you have monitoring (e.g. Nagios) tracking thinks like server CPU and memory use? If so, does that show anything? (I am guessing from what you say that you do have monitoring like this, and it shows nothing.)
Bottom line, this code almost always works reliably for many people around the world, and it has not changed in quite a long time. It would be quite surprising if there was a bug in it.