We consider a pattern of neutrino masses in which there is an approximate mass degeneracy between the two mass eigenstates most coupled to the νμ and ντ flavour eigenstates. Earth-matter effects can lift this degeneracy and induce an effectively maximal mixing between these two generations. This occurs if νe's contain comparable admixtures of the degenerate eigenstates, even rather small ones. This provides an explanation of the atmospheric neutrino anomaly in which the {\it ab initio} introduction of a large mixing angle is not required. To test this possibility we perform a novel and detailed analysis of the 52 kiloton-year SuperKamiokande data, and we find that in a large region of parameter space the corresponding confidence levels are excellent. The most recent results from the Chooz reactor experiment, however, severely curtail this region, so that the conventional scenario with nearly maximal mixing angles --which we also analyse in detail-- is supported by the data.