A number of prominent epistemologists claim that the principle of sensitivity “play[s] a starring role in the solution to some important epistemological problems ” (DeRose 2010: 161; also Nozick (1981)). I argue that traditional sensitivity accounts fail to explain even the most basic data that are usually considered to constitute their primary mo-tivation. To establish this result I develop Gettier and lottery cases involving necessary truths. Since beliefs in necessary truths are sen-sitive by default, the resulting cases give rise to a serious explanatory problem for the defenders of sensitivity accounts. It is furthermore argued that attempts to modally strengthen traditional sensitivity ac-counts to avoid the problem must appeal to a n...