High-resolution N-body simulations of galactic cold dark matter haloes indicate that we should expect to find a few satellite galaxies around the Milky Way whose haloes have a maximum circular velocity in excess of 40 km s−1. Yet, with the exception of the Magellanic Clouds and the Sagittarius dwarf, which likely reside in subhaloes with significantly larger velocities than this, the bright satellites of the Milky Way all appear to reside in subhaloes with maximum circular velocities below 40 km s−1. As recently highlighted by Boylan-Kolchin et al., this discrepancy implies that the majority of the most massive subhaloes within a cold dark matter galactic halo are too concentrated to be consistent with the kinematic data for the bright Milky Way satellites. Here we show that no such discrepancy exists if haloes are made of warm rather than cold dark matter because these haloes are less concentrated on account of their typically later formation epochs. Warm dark matter is one of several possible explanations for the observed kinematics of the satellites.