I have a Dell Inspiron 500M notebook that has a mind of its own.
If my note is shutdown over a period of 24 hours, it will not boot up unless of course I waited long enough to the notebook to charge, even though the laptop is physically plugged into a wall socket. Regardless, the electrical cord must be connected and charge for a while.
Otherwise, if there is a docking station nearby, I will have no problems booting it up right away. This has never happened to me before in the last 4 years. Can someone tell me if this has anything to do with the CMOS battery that is failing or if it is an issue with the power on button on the notebook.

What happens if you remove the battery and then try to shut down and start up on AC power? I wouldn't believe this to be as cmos battery issue since the problem isn't uniform across all configurations.

it sounds like the system battery has a problem...try with a battery known to work properly...