MySQL Memory & max_connections Performance Tuning
After thorough research, we determined that the problem was that WordPress was being attacked and it was causing MySQL to use the maximum amount of memory allotted to it in the configuration files.  Even caching of WordPress pages could not stop this illegitimate traffic from using all of the memory because of the sheer volume of IP addresses the attackers had at their disposal.  We ended up getting CloudFlare and that seems to have resolved the problems by blocking the DDoS at the DNS level. We will also be looking at getting more memory for the server, to prevent this from happening with heavy legitimate traffic in the future.