I have a Dell Vostro 1400 and two out of the 3 blue lights that look like 'locks' RIGHT next to the on button are flashing when I try to turn on.  THey flash for a moment and then the computer turns off.  I've checked battery LED lights and they are all on.  I've plugged it in and the same happens.  Can you help?Hope you have a backup.
Unplug power.
Remove battery.
Wait about 3 min.

Put in battery, plug in, hold power button in for five seconds and RELEASE. And wait. It should restart.

BTW, any USB devices you have should be unplugged for this to work right.
1. How long have you owned the machine and is it still under warranty?
2. Did it happen just suddenly or any previous events, changes before the problem?
3. Are you familiar in opening up laptop casing, just in case?Ok.  I have tried to take battery out but that didn't work.  But, I did not hold the button down 5 secs.  I'll try that. 

I have owned it for about 4 years, but it's been working great til now. 

For the past two weeks or so, I had to keep pushing the on button and eventually it would boot up, but not anymore.  Not under warranty.

Not familiar with opening laptop casing.....what to do?

Thanks guys!Have you ever replaced the battery?
In some cases a dead battery will stop the UNIT from working.
I think the Dell Vostro 1400 can OPERATE with no battery in place.
Not sure of that. Some say never do that. But the general consensus is that there is no harm.
Replace coin cell battery:  http://support.dell.com/support/edocs/systems/ins1420/en/SM/coinbatt.htm#wp1106292