A friend and I have identical hardware setups consisting of a duemilanove and the standard ethernet shield. Without getting into the code, we've noticed that the device will stop responding to pings, web server requests, etc. after brief power outages - the kind that last a tenth of a second and reset half the stuff in your house but not the rest. I'm not sure if the Arduino itself is resetting but I think not - it seems to be riding out the outage and continuing to process like nothing happened. If the power is out long enough to reset the Arduino then everything is fine. I'm thinking the enet shield might be losing its settings and they don't get refreshed because the arduino itself doesn't go through setup() again.
Anybody see anything like this before? Any creative ideas of what to try? I'm wondering if I might do something on the ethernet shield's power supply to help it ride out flickers of power - or if there's a way the arduino itself can know that it's unable to talk to the enet shield and do something about it.