This recently happened to my father's Samsung LCD TV. When he would go to turn it on it would click but not turn on. Often, if he kept trying, it would eventually turn on. Once it was powered on it would work fine. He could then turn it off and if he tried to turn it on within a few minutes, it would turn back on. If he waited a while and the internals cooled down, it would not turn back on without multiple tries.
He had a tech come out and the problem was bad capacitors. The tech demonstrated that the capacitors were 'popped' and the tops were pushed out and a small amount of gel was exposed. Upon replacing the bad capacitors the tv now turns on properly.