The discovery of mirror neurons in macaques and of a similar system in humans has provided a new and fertile neurobiological ground for rooting a variety of cognitive faculties. Automatic sensorimotor resonance has been invoked as the key elementary process accounting for disparate (dys)functions, like imitation, ideomotor apraxia, autism, and schizophrenia. In this paper, we provide a critical appraisal of three of these claims that deal with the relationship between language and the motor system. Does language comprehension require the motor system? Was there an evolutionary switch from manual gestures to speech as the primary mode of language? Is human communication explained by automatic sensorimotor resonances? A positive answer to these questions would open the tantalizing possibility of bringing language and human communication within the fold of the motor system. We argue that the available empirical evidence does not appear to support these claims, and their theoretical scope fails to account for some crucial features of the phenomena they are supposed to explain. Without denying the enormous importance of the discovery of mirror neurons, we highlight the limits of their explanatory power for understanding language and communication.