A molecular-scale description of water and ice is important in fields as diverse as atmospheric chemistry, astrophysics, and biology. Despite a detailed understanding of water and ice structures on a multitude of surfaces, relatively little is known about the kinetics of water motion on surfaces. Here, we report a detailed study on the diffusion of water monomers and the formation and diffusion of water dimers through a combination of time-lapse low-temperature scanning tunnelling microscopy experiments and first-principles electronic structure calculations on the atomically flat Cu(111) surface. On the basis of an unprecedented long-time study of individual water monomers and dimers over days, we establish rates and mechanisms of water monomer and dimer diffusion. Interestingly, we find that the monomer and the dimer diffusion barriers are similar, despite the significantly larger adsorption energy of the dimer. This is thus a violation of the rule of thumb that relates diffusion barriers to adsorption energies, an effect that arises because of the directional and flexible hydrogen bond within the dimer. This flexibility during diffusion should also be relevant for larger water clusters and other hydrogen-bonded adsorbates. Our study stresses that a molecular-scale understanding of the initial stages of ice nanocluster formation is not possible on the basis of static structure investigations alone.