It looks like OOM Killer [out of memory killer] was going wild over a few accounts - and the amount it was running was causing system wide issues. The system itself didn't exhaust RAM, but the user-level issues did cause some problems.
We adjusted some system settings as per CloudLinux's advice and we believe this issue should not recur. We are going to continue watching the server closely regardless.
The changes recommended by CloudLinux have been applied to all servers.