Sheldon Silver has long run New York even though he is the leader of only one house, the State Assembly. He sits back, says nothing, but will not pass a budget or any other bill until it fits his demands. He lets the governor stand in the limelight every day giving updates and explaining his position and eventually the voters blame the governor for not getting the job done. There is always a perception that the chief executive is responsible for whatever happens on his watch, see: Bush and the sub-prime mortgage crisis.
If Boehner and the Republicans cave on the No Tax Rate Increases pledge they cease to have a reason to exist.