This is a Dell PowerEdge SC1420. No problem with this server until a brief power outage. Now, it restarts fine, but when it is shut down, either intentionally or not, the 4 LEDs on the front of the server blink very dimly but it won't turn on unless I unplug the AC, open the case, remove and re-install or replace the coin battery. Then it will boot fine. I've replaced the coin battery with a new one but the problem still occurs after a shutdown. The server is out of warranty.
Having replaced the CMOS battery, verify the BIOS settings for what to do after a power failure (usual options are "off", "last state" and "on" -- You probably want "on").
Failing that, hit the Dell website, download a copy of the diagnostic CD appropriate to your server and see what it says.
No matter what you do, make sure your backups are current and good (i.e. "You can restore the system from the backups without needing the original system to be functional"). Also plan on replacing the server - If it's out of warranty and acting up it will fail at the least convenient time.