Ok, first off, I have a Dell Latitude D820, and it was working just fine one day. I turned it off and it shut down just fine. When I went to turn it back on the next day, I pressed the power button, and the Numlock, Scroll Lock, and Caps lock leds came on, but there was no noise at all, no hd spinning, no fan running, and nothing on the screen. Then after about 4 seconds it turned itself back off. It does this every time I try, and doesn't matter if its running on battery or AC power.
Welcome to the community. Sounds like its a Hardware problem. Try reseating battery, drives and memory(ram). Then return devices and try booting again. If same thing happens, possible motherboard failure. Dell can replace it as long as its under warranty.
Hope this helps. Thank you,