Under favorable conditions, many proteins can assemble into macroscopically large aggregates such as the amyloid fibrils that are associated with Alzheimer's, Parkinson's, and other neurological and systemic diseases. The overall process of protein aggregation is characterized by initial lag time during which no detectable aggregation occurs in the solution and by maximal aggregation rate at which the dissolved protein converts into aggregates. In this study, the correlation between the lag time and the maximal rate of protein aggregation is analyzed. It is found that the product of these two quantities depends on a single numerical parameter, the kinetic index of the curve quantifying the time evolution of the fraction of protein aggregated. As this index depends relatively little on the conditions and/or system studied, our finding provides insight into why for many experiments the values of the product of the lag time and the maximal aggregation rate are often equal or quite close to each other. It is shown how the kinetic index is related to a basic kinetic parameter of a recently proposed theory of protein aggregation.