I've been through the server logs (access logs for the site, and error logs for the software). There is nothing showing any errors at the time, and it seems that the issue only happens to one person at a time. If it was an issue with communication with Cloudflare and the server, everyone accessing the site would be getting the errors.
OK, so I think I know what this specific issue was. I was sending a backup of the forum files to Amazon S3. The problem is that is can use the whole network bandwidth, so I need to put a rate limit on sending stuff to S3, as it will use the while Gigabit port.