What kind of connection? Wifi or ethernet? It PROBABLY doesn't make a difference, but bad airport cards have been known to cause kernel panics.

The first thing I'd suggest though is updating to 10.6.8 unless you have a specific reason for staying at .7. A lot of times Apple includes fixes for things in point releases they don't bother mentioning in the changelog. So there's at least a possibility that your issue was resolved in 10.6.8. If you ever plan to upgrade to 10.7, it'd also be a good idea to have 10.6.8 installed.

The other possibility that springs to mind for your issue however, is bad RAM. Usually the symptoms are a bit less specific, but bad RAM can have a number of odd symptoms. If you run AHT off of the Applications DVD that came with your unit and tick off the extended test box, it will do a simple memory test on your RAM.