My ThinkPad X1 refuses to start, and pressing on/off buttons do not work.
Near the slice battery power port and also near the big rubber foot at the rear, is a tiny hole. Having had to reset my GPS unit countless times with paper clip, I wondered if this was a reset switch hole as well. I very gently inserted a straightened paper clip into this hole and felt the familiar click of hitting a reset switch. I turned my X1 over and it powered up normally.
The ThinkPad X1 has an integrated battery within the laptop, which may prevent the on/off button trick from resetting the ThinkPad as it usually does in many other ThinkPad models. To activate the reset switch, just push on the pin and you will feel some resistance and small click noise, don't push too hard as this may damage the reset switch mechanism.