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This paper extends very recent results on discrete- time nonlinear fault detection and isolation to the case of discrete-time nonlinear systems with unstructured modeling uncertainty and partial state measurement. The fault diagnosis architecture consists of a fault detection and approximation estimator and a bank of fault isolation estimators, each corresponding to a particular type of fault. A time-varying threshold that guarantees no false-positive alarms and fault detectability conditions are derived analytically. For the fault isolation scheme, we design adaptive residual thresholds associated with each isolation estimator and obtain sufficient conditions for fault isolability. To illustrate the theoretical results, a simulation example based on a input-output discrete-time version of the three-tank benchmark problem is presented.