LInux: How do I diagnose / isolate what's causing "random" hangs and spontaneous reboots?

Problem

(originally posted on serverfault) So, rather than guessing just what the cause is (though my money's on the nvidia drivers), where do I start looking to pin down some facts? I've been through /var/log on several occasions but there's a LOT of stuff in there and I can't (yet) spot the important bit…