Thanks everyone, I'm looking into it.
The website has been the target of a number of DDoS (denial of service) attacks off and on for the last several months. I've blocked many IP addresses to try to ease the burden on the server, but it doesn't seem to make much difference for very long.
And to make things worse, the hosting company I use is not being very helpful. They want to put the blame on me even though there is very little I can do to stop a DDoS attack. So I suspect these 508 "Resource Limit" errors are actually due to my host putting limits on the website and not because there is an attack happening.
I might switch to another host if this continues, but that doesn't guaranty the problem will go away.
I apologize for the inconvenience, I'm going to keep doing what I can to resolve the problem.