Beware Rogue Passenger Orphans!
One other strange symptom was that every once in a while when you'd visit the site in a browser, it wouldn't have any styling. Muy raro.
After a good amount of digging into this issue, it finally became clear what was happening.
While looking through the log files to try to figure out what was going on, I ran across the superbly named "passenger-orphan-killer.log". My eye being drawn to this, I decided to peruse it.
This turned out to hold the missing clues as to what was happening. For the past few days, it was filling up with messages such as this:
Tue Oct 4 03:00:01 EDT 2011
Killed 1 orphaned ruby process(es): 21901
Thu Oct 17 03:00:01 EDT 2013
/usr/local/bin/passenger-orphan-killer: line 14: /opt/ruby-enterprise-1.8.7-2011.03/lib/ruby/gems/1.8/gems/passenger-3.0.7/bin/passenger-status: No such file or directory
passenger-status failed. aborting
Fri Oct 18 03:00:01 EDT 2013
It was trying to make a call to passenger-3.0.7/bin/passenger-status, which didn't exist! Passenger must have been upgraded, as passenger-3.0.8/bin/passenger-status did exist.
Once we'd successfully executed the Passenger orphan, and gotten the Passenger Orphan Killer working once more, everything was shipshape.