Hi, thank you for contacting JustAnswer.com. My name is Russell. I will do my best to provide the right answer to your question.
Your Thinkpad T410 seems dead, I take it - no fan activity or whirr, no lights on keyboard, no power light.
Try removing the battery - if that model has a removable power battery - and run it only on the power cord. Don't hold down the power button, just give it a firm push for 1 second. Holding it down continuously will shut the machine down, not turn it On.
Check the power cord or AC Adapter, also. Is it plugged into a good outlet? test the outlet. Make sure the GFI switch of the outlet is in the right position, if it has a GFI button. (Usually those are only found in bathroom outlets.)
Test the outlet with another device, too.
And, inspect the cord, and where the cord plugs in, to see if it has any visible flaws.
How old is your laptop? has it ever been dropped, or used in very hot (above 85 degrees) situations?