I'm assuming the system worked fine until recently.
Did you install any new software recently, particularly file system utils?
Check for viruses. This sounds like the most likely cause to me. I've had that same problem caused by viruses on W9X systems, but I've been lucky and not encountered any virus problems with NT4.
Have you run diagnostics on the RAM? Not the boot-up mem test, but a good diagnostic util like Norton or a shareware util (check out download.com.com). Run diags on the whole system, but remember that without loopback plugs the serial & parallel ports won't respond well. Your Dell may have come with utils that might be handy too. Check out the docs. Otherwise, do you have one 512MB module, or 2x256, etc? If multiple modules, try taking out one at a time to see if the problem goes away.
Not as likely: IRQ and IO conflicts can cause all sorts of nasty problems. Make sure your resources aren't conflicting between video and NIC, mouse and modem, etc.