It looks like we ran out of both RAM and swap space causing the system to completely lock up. This was due to a combination of factors, I think--for one, we'd been running pretty low on RAM recently. Second, the nightly backup process kicked in, running gzip on some large (GB+) files. Finally, memcached seems to have run up to about 280MB of memory usage (despite my attempt at configuring it to us only 64MB).
At any rate, these stresses caused some serious strain on the overall system by, I believe, slowing down DB access considerably, resulting in the DB maxing out the number of connections available. This killed the C Board and also led to a ton of open HTTP connections. I suspect that some combination of all of these activities created a cascading failure that forced a server reboot.
It isn't clear if this would have happened had vBulletin 3 still been installed (either if it was caused memcached pushing the memory limits over the top or more processing required for vBulleitn 4), but based on what I was seeing in top, I suspect that adding additional RAM to the system is a good step to solving tis problem. We're currently running with only 2GB of RAM and a 2GB swap.