Amongst philosophers and cognitive scientists, modularity remains a popular choice for an architecture of the human mind, primarily because of the supposed explanatory value of this approach. Modular architectures can vary both with respect to the strength of the notion of modularity and the scope of the modularity of mind. We propose a dilemma for modular architectures, no matter how these architectures vary along these two dimensions. First, if a modular architecture commits to the informational encapsulation of modules, as it is the case for modularity theories of perception, then modules are on this account impenetrable. However, we argue that there are genuine cases of the cognitive penetrability of perception and that these cases challenge any strong, encapsulated modular architecture of perception. Second, many recent massive modularity theories weaken the strength of the notion of module, while broadening the scope of modularity. These theories do not require any robust informational encapsulation, and thus avoid the incompatibility with cognitive penetrability. However, the weakened commitment to informational encapsulation greatly weakens the explanatory force of the theory and, ultimately, is conceptually at odds with the core of modularity.