This looks like the server becoming overloaded, rather than a PHP error. So, rather than looking in the logs, you need to be monitoring the server performance when the problem happens. Which resource is getting used up? It could be CPU, memory, disc i/o.
Once you know where the bottle-neck is, you can tune your server to make it perform better. The quiz should be able to handle much more than a couple of dozen simultaneous users, if the server is tuned reasonably well.