Cryogenic transmission electron microscopy and small-angle neutron scattering recently have revealed that the well-known thermoreversible gelation of methylcellulose (MC) in water is due to the formation of fibrils, with a diameter of 15 ± 2 nm. Here we report that both the linear and nonlinear viscoelastic response of MC solutions and gels can be described by a filament-based mechanical model. In particular, large-amplitude oscillatory shear experiments show that aqueous MC materials transition from shear thinning to shear thickening behavior at the gelation temperature. The critical stress at which MC gels depart from the linear viscoelastic regime and begin to stiffen is well predicted from the filament model over a concentration range of 0.18-2.0 wt %. These predictions are based on fibril densities and persistence lengths obtained experimentally from neutron scattering, combined with cross-link spacings inferred from the gel modulus via the same model.