Having concepts is a distinctive sort of cognitive capacity. One thing that conceptual thought requires is obeying the Generality Constraint: concepts ought to be freely recombinable with other concepts to form novel thoughts, independent of what they are concepts of. Having concepts, then, constrains cognitive architecture in interesting ways. In recent years, spurred on by the rise of evolutionary psychology, massively modular models of the mind have gained prominence. I argue that these architectures are incapable of satisfying the Generality Constraint, and hence incapable of underpinning conceptual thought. I develop this argument with respect to two well-articulated proposals, due to Dan Sperber and Peter Carruthers. Neither proposal gives us a satisfactory explanation of Generality within the confines of a genuinely modular architecture. Massively modular minds may display considerable behavioral and cognitive flexibility, but not humanlike conceptualized thought.