The well-known Fermi-Pasta-Ulam (FPU) phenomenon (lack of attainment of equipartition of the mode energies at low energies for some exceptional initial data) suggests that the FPU model does not have the mixing property at low energies. We give numerical indications that this is actually the case. This we show by computing orbits for sets of initial data of full measure, sampled out from the microcanonical ensemble by standard Monte Carlo techniques. Mixing is tested by looking at the decay of the autocorrelations of the mode energies, and it is found that the high-frequency modes have autocorrelations that tend instead to positive values. Indications are given that such a nonmixing property survives in the thermodynamic limit. It is left as an open problem whether mixing actually occurs, i.e., whether the autocorrelations vanish as time tends to infinity.