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This paper extends very recent results on a distributed fault diagnosis methodology for nonlinear uncertain large-scale discrete-time dynamical systems to the case of partial state measurement. The large scale system being monitored is modeled, following a divide et impera approach, as the interconnection of several subsystems that are allowed to overlap sharing some state components. Each subsystem has its own Local Fault Diagnoser: the local detection is based on the knowledge of the local subsystem dynamic model and of an adaptive approximation of the interconnection with neighboring subsystems. A consensus-based estimator is used in order to improve the detectability of faults affecting variables shared among different subsystems. Time-varying threshold functions guaranteeing no false-positive alarms and analytical fault detectability sufficient conditions are presented as well.