I had similar problems w/my HP a couple years ago...
I had taken it apart to put some arctic silver on the heatsink and didn't completely reseat the fan connector. It'd last about that same amount of time...sometimes it'd start booting the OS and then fail. The issue was that it was simply overheating and shutting itself down.
Pop it open, check the heatsink/CPU thermal compound and make sure all the fans are free spinning. If the fans are stuck on dust then remove them, remove the stickers or rubber plugs that cover the axles, and put a tiny drop of oil in there while spinning the blades by hand. If it's really gummed up then blow the first round or two out with canned air and then finish it up with a clean shot.
Other than that I'd attempt re-flashing the BIOS if you can get it to let you. Not all laptops have an emergency reflash jumper like desktops, so you may be SOL if you can't get it to boot off a floppy. You should be able to find flash and emergency flash recovery info on Dell's site along with the firmware image.