System freezes. Can't find anything in the logs

Sounds like a potential hard drive problem. The behavior you describe is exactly what you would see if the hard drive suddenly stopped responding to requests, or started returning errors. Try running smartctl -a /dev/sda, where sda is the device of your hard drive, to see if the hard drive has logged any errors.

You would see an explanation of the error in the kernel's dmesg output. Of course, if you can't login, you can't run dmesg. However, if you log in first and run the command on the text console, it may still be in memory so that you could run it again after the system has crashed.

The other option would be to use the netconsole feature of Linux to send the debugging information directly to another machine instead of writing it to disk. This is a little tricky to set up and requires two machines, but is a good thing to try if everything else fails.