Water plays a major role in biosystems, greatly contributing to determine their structure, stability, and function. It is well known, for instance, that proteins require a minimum amount of water to be fully functional. Despite many years of intensive research, however, the detailed nature of protein-hydration water interactions is still partly unknown. The widely accepted "protein dynamical transition" scenario is based on perfect coupling between the dynamics of proteins and that of their hydration water, which has never been probed in depth experimentally. I present here high-resolution elastic neutron scattering measurements of the atomistic dynamics of lysozyme in water. The results show for the first time that the dynamics of proteins and of their hydration water are actually decoupled at low temperatures. This important result challenges the "protein dynamical transition" scenario and requires a new model to link protein dynamics to the dynamics of its hydration water.