A new method has been applied to the determination of neutron nonelastic cross sections for iron 56Fe and lead 208Pb for energies between 5 and 26 MeV. These data have estimated errors of only a few percent and do not suffer from the ambiguities encountered in earlier nonelastic data. We attempt to fit these high-precision data using both a semiclassical single phase shift model (nuclear Ramsauer model) as well as a recent global optical model that well reproduces a wide body of neutron scattering observables. At the 5% uncertainty level, both models produce satisfactory fits. However, neither model gives satisfactory fits to these new precise data. We conclude that fitting precise data, i.e., data with errors of ~2% or less, may require a nuclear mass dependence of radii that reflects structure effects such as shell closures.