On our old server, our hosting company said the server was running out of RAM and then became unresponsive. I haven't checked about the new server yet.
Unresponsive how? Can you ssh to it? Can you log to Postgres?
For how long does it happen? Till you reboot?
If it is a server crash then that is not a normal behavior and you should check your hardware. An exhaustive memory test is recommended.
Is there any kind of diagnostics you can think of that would help get to the root of the problem - something I could put in a cron job or a monitor app I could run on the server that would at least tell us what is going on if / when it happens again?
Increase logging on PostgreSQL. Especially log checkpoints and locks.
While experiencing the problem and if you are able to log to the server, a vmstat 1 10 will tell you what is going on with your I/O system in a 10 second span.