Primatologists generally agree that monkeys lack higher-order intentional capacities related to theory of mind. Yet the discovery of the so-called “mirror neurons” in monkeys suggests to many neuroscientists that they have the rudiments of intentional understanding. Given a standard philosophical view about intentional understanding, which requires higher-order intentionality, a paradox arises. Different ways of resolving the paradox are assessed, using evidence from neural, cognitive, and behavioral studies of humans and monkeys. A decisive resolution to the paradox requires substantial additional empirical work and perhaps a rejection of the standard philosophical view.