One of my wordpress installs has been under stress lately, it goes through periods of unreachability, sometimes daily. This site seems like it's an attractant for prodders and hackers (mostly idiotic: wp-login usename admin password admin attempts). When complaining to the support staff, I've received several boilerplate suggestions about tightening up wordpress and memory utilization. I've been working on tightening up wordpress for over 18 months and have gotten pretty good at it.
I researched in depth on my own today and found an extraordinary correlation with wp-login hackers and memory overruns. I'm getting 10-20 wp-loing requests in a few seconds of each other.
The suggestion I get from DH support is always the same: here is some nerd info on how to use less memory for wordpress, but you could solve everything by upgrading to VPS.
In this case, I think the appropriate solution was to more aggressively control who can get to start wp-login. I never saw this suggestion from your troubleshooting staff, and I suspect if I were on a VPS I sumply may allow more simultaneous hackers to mess with my site.
Summary: Have your troubleshooting staff spend 2 minutes and look at BOTH the access log and the error log on the complaining site and correlate what was happening when memory issues exposed themselves.