Hiya, I have been experiencing the same problem I think. My T420 will not respond to just pressing the power button on. It would only boot up if I plugged the charger in , as if it needed a slight surge of power. I later found out that if I pressed the Esc button and held it down at the instant I pressed the power button, it will then start up as normal, on the battery only. Have tried this several times so far which seems to work for me, unusual though it sounds. Not sure exactly what’s going on but appears to work. Give it a go, and Good Luck.