In this paper, the state estimation problem is investigated for a class of discrete nonlinear systems via sensor networks. A novel robust distributed extended Kalman filter, which can handle norm-bounded uncertainties in both the system model and its Taylor series expansion, is developed with ensured estimation performance. The filter is distributed in the sense that for each sensor, only its own measurements and its neighbors' information are utilized to optimize the upper bound of the estimation error covariance. Besides, a sufficient condition for the proposed algorithm is derived, which is simple and user-friendly since it depends on the property of the original nonlinear system instead of the estimation error covariance calculated at every step. Finally, the simulation results are presented to demonstrate the effectiveness of the filtering algorithm.