The growth of cloud droplets by diffusion of water vapor in a three-dimensional homogeneous isotropic turbulent flow is considered. Within a simple model of advection and condensation, the dynamics and growth of millions of droplets are integrated. A droplet-size spectra broadening is obtained and it is shown to increase with the Reynolds number of turbulence by means of two series of direct numerical simulations at increasing resolution. This is a key point toward a proper evaluation of the effects of turbulence for condensation in warm clouds, where the Reynolds numbers typically achieve extreme values. The obtained droplet spectral broadening as a function of the Reynolds number is shown to be consistent with dimensional arguments. A generalization of this expectation to Reynolds numbers not accessible by direct numerical simulation (DNS) is proposed, yielding upper and lower bounds to the actual size spectra broadening. It is argued that the lower bound is the relevant limit at high Reynolds numbers. A further DNS matching the large scales of the system suggests consistency of the picture drawn. The assumptions underlying the model are expected to hold up to spatial scales on the order of 100 m; no direct comparison with in situ measures is possible. Additional effort is needed to evaluate the impact of this effect for condensation in more realistic cloud conditions.