We have performed high precision measurements of the temperature dependence of the magnetic penetration depth λ(T) in pure aluminum and cadmium samples. These superconductors and most of the other superconducting metals are predicted to be in the nonlocal limit of electrodynamics. However, most previous experimental works on λ(T) in Al reported large deviations from the expected nonlocal behavior. Here we present experimental data which agree remarkably well with theory, giving very strong evidence that nonlocal superconductivity indeed occurs in both Al and Cd. More strikingly, the cadmium data are consistent with the existence of the crossover from nonlocal to local behavior predicted to occur in all superconductors near Tc.