We have reconsidered the atmospheric neutrino anomaly in light of the laetst data from Super-Kamiokande contained events and from Super-Kamiokande and MACRO up-going muons. We have reanalysed the proposed solution to the atmospheric neutrino anomaly in terms of non-standard neutrino-matter interactions (NSI) as well as the standard nu_mu -> nu_tau oscillations (OSC). Our statistical analysis shows that a pure NSI mechanism is now ruled out at 99%, while the standard nu_mu -> nu_tau OSC mechanism provides a quite remarkably good description of the anomaly. We therefore study an extended mechanism of neutrino propagation which combines both oscillation and non-standard neutrino-matter interactions, in order to derive limits on flavour-changing (FC) and non-universal (NU) neutrino interactions. We obtain that the off-diagonal flavour-changing neutrino parameter epsilon and the diagonal non-universality neutrino parameter epsilon' are confined to -0.03 < epsilon < 0.02 and |epsilon'| < 0.05 at 99.73% CL. These limits are model independent and they are obtained from pure neutrino-physics processes. The stability of the neutrino oscillation solution to the atmospheric neutrino anomaly against the presence of non-standard neutrino interactions establishes the robustness of the near-maximal atmospheric mixing and massive-neutrino hypothesis. The best agreement with the data is obtained for Delta_m^2 = 2.3*10^{-3} eV^2, sin^2(2*theta) = 1, epsilon = 6.7*10^{-3} and epsilon' = 1.1*10^{-3}, although the chi^2 function is quite flat in the epsilon and epsilon' directions for epsilon, epsilon' -> 0.