This article studies the problem of fault detection and isolation (FDI) for a class of uncertain nonlinear systems via a residual signal generated by a novel nonlinear adaptive observer. The considered faults are modelled by a set of time-varying vectors, in which a prescribed subset of faults are specially monitored and thus separable from the other faults. In the presence of Lipschitz-like nonlinearities and modelling uncertainties, the sensitivity of the residual signal to the monitored faults and its insensitivity to the other faults are rigorously analysed. Under a persistent excitation condition, the performances of the proposed fault diagnosis scheme, including the robustness to uncertainties, the quickness of estimation, the accuracy of estimation, the sensitivity to the monitored faults and the insensitivity to the complement faults, are quantified by a series of explicit design functions relevant to the observer parameters. It turns out that the number of faults which can be completely diagnosed is independent of the number of output sensors. A simulation example is given to illustrate the effectiveness of the proposed FDI method.