Protein aggregation is an important biophysical phenomenon, and it is technically challenging to quantify. Scattering studies in concentrated protein solutions are not in complete agreement over the existence of an equilibrium cluster phase. We use pulsed-field-gradient NMR spectroscopy to characterize diffusion in the long-time limit in concentrated lysozyme solutions and find strong evidence for the existence a an equilibrium phase that consists of both lysozyme monomers and clusters (aggregates). They indicate too that there is rapid exchange between monomer and aggregate on the NMR time scale, and that macroscopic measurables (e.g., the relaxation rate and the observed diffusion coefficient) reflect a weighted average of the two fractions. Our results are quantitatively compared, with no fit parameters, to simple theories of macromolecular crowding.