Turn On Twice To Start Laptop

I've got something going on with my Gateway GWTN156-1GR laptop.

If I shut down the laptop, and leave the power plugged in over night, when I start the laptop up the next morning it'll start right up with a single push of the power button.

But, if I shut the laptop down, with a full battery charge, and unplug the power supply, when I try to restart the laptop top the next morning I have to push the power button a first time, which the power light, etc comes on, but the computer doesn't do anything else at all. Then, I have to push and hold the power button to shut it off, and push it again to re-start it and it'll start up. AND, when I get into windows the battery level is still showing fully charged.

I've pretty much tried everything I could find online: the power settings, bios settings, etc and nothing changes it.

Any idea's on what might be causing this?

I'm beginning to wonder if the memory isn't acting up, but it's onboard memory so I can't just change out a stick of memory to check it.