Intermittently doesn't turn on?

I have two televisions in my house that both experinced the same symptoms.
Please excuse me as I dont have the make and models of either on me right now.One day the larger TV wouldn't turn on. We pushed the power on button over and over again, so I went out and bought a new TV. When we were about the throw it out, we plugged it in and tried it one more time..the stupid thing turned on..and hasent had a problem since.
Then i went out and bought a used 13" for my sons room. The Exact same thing happened to this one, but because of the experience with the larger one, we waited a while and it turned on. Now Im afraid to shut either Unit off. Can anyone offer any suggestions? And all the power outlets in my house work great and wouldnt turn on with the remote control or the power switch.
Thanks