Dell laptops tend to exhibit this behavior when their processors are not seated correctly. Dell has been using a ZIF (Zero Insertion Force) socket lately that uses a cam-screw to lock the pins in the socket. Sometimes these cams are not fully locked down and work loose, causing the pins to disconnect from the socket's contacts. If your processor is one of the newer Core2 chips without actual pins on it, it won't have this arrangement, but a simple lever/frame locking arrangement. These don't generally work loose, in which case it could be that the processor is overheating or may have burnt out.
The troubleshooting steps I would try are as follows:
Try booting with various power configurations, i.e. battery and AC adapter connected, then battery alone, then AC adapter alone. If the symptoms are the same no matter how you boot, you rule out power as a cause.
Next, I'd pop the covers on the various user-accessible cards, such as WiFi, modem, memory etc. and verify that all are seated properly.
Try booting with various memory configurations. Try the cards one at a time (if you have 2) and in different slots. Then try booting without any memory in the computer to see if it behaves differently. If it does, you may have a bad memory stick.
After that, if you still have the same symptoms, it's probably some combination of processor / motherboard / video card. If the video card is shot, it COULD cause the computer to do this. If it's an integrated card, it's new motherboard time. Being an XPS, it's likely not though.
If you feel comfortable enough with removing the keyboard, you can carefully take a look inside and make sure everything is secure (cables, cards etc.)
I'm putting my money on the processor either being loose or defective. Unfortunately, without being able to boot at all, you can't even run the diags, so this one will be hard to diagnose.
Do you allow adequate cooling while using your XPS?