The aim of this chapter is to investigate the possible connections between deeply opaque ignorance and incommensurability (I will later explain both concepts in detail). More specifically, does incommensurability imply deeply opaque ignorance? The answer to this question will shed some light on our epistemic situation in general—namely, whether deeply opaque ignorance is the situation in which we normally find ourselves in the sciences. I should mention at the outset that this is a theoretical philosophy of science chapter. It therefore does not belong to critical political philosophy of science (from where much of recent agnotology research springs), nor to logic or epistemology. I will first explicate my core concepts, opaque ignorance and deeply opaque ignorance, and then incommensurability and strong incommensurability. In a second step, I will formulate a hypothesis about the connection between these concepts. In a third step, I will present three case studies from the history of science, one from physics and two from medicine, in order to illustrate the concepts and make the hypotheses plausible. In physics, I will discuss the situation of classical mechanics and precession of the perihelion of Mercury, and in medicine I will discuss the emergence of virus research, on the one hand, and prion research, on the other. In the end, I will draw some tentative conclusions about our epistemic situation in the sciences in general.