This paper develops an adaptive approximation based approach for distributed fault diagnosis for a class of interconnected continuous-time nonlinear systems with modeling uncertainties and measurement noise. The proposed approach integrates learning with filtering techniques and allows the derivation of tight detection thresholds. This is accomplished in two ways: at first by learning the modeling uncertainty through adaptive approximation methods, so that the learned function is used for the derivation of the residual signal, and then by using filtering for dampening measurement noise. The required signals for both tasks are derived through a two-stage filtering process, by exploiting the properties of the filtering framework. Finally, simulation results are used to demonstrate the effectiveness of the proposed approach