I think you may need to check the RAM. Snow Leopard will be more demanding and may expose pre-existing problems.
To test the memory, get memtest and run it in single user mode, where it will test as much memory as possible, more than with the OS loaded.
You can get memtest + directions from the link below. However, ignore running it from Terminal. Instead, boot into SU Mode, Cmd-s at the startup chime. (Best to startup from a full shutdown.)
At the prompt, simply type /usr/bin/memtest all 3 -L (From this link It will be installed in /usr/bin/) Then hit return. This will run three loops of memtest and create a log in Console in Utilties.
If you want to run memtest longer, which may be advisable, since RAM errors can be very elusive, you can change the number of loops it runs by changing the number. For example, to run 8 loops you would enter /usr/bin/memtest all 8 -L.
If you want to quit the test, just hit control-c
When finished, you can just type in "reboot" and hit return.
http://osxdaily.com/2011/05/03/memtest-mac-ram-test/
Direct link for the download.
http://cdn.command-tab.com/2008/memtest_422.zip