Several things:
1. You need to set up Debugging. Set it to level at least ALL, and preferably DEVELOPER. Leave 'Display debug messages' turned off, so the messages are just written to the server log files, but find those logs and see what messages are being written there, if any.
2. Also look at the non-error logs, that is the web server logs that list every request made, and the Moodle logs in the database. What else is going on on the server when students experience problems?
3. "We have tried duplicating the students' experience" is absolutely the right thing to try to do, if you can. You need to look for any common factors among the students who are having problems (and differences from students who are not). Things that might be relevant: Student's web browser. Details of the students' roles (Try Admin -> Check permissions in the quiz). ... Those are the two most obvious, but there are other potential similarities or differences (groups, time zones, ...)
4. As well as PHP errors, can you monitor other aspects of the server performance. E.g. CPU load and memory usage. Anything suspicious there?