This paper introduces a novel cognitive fault diagnosis system (FDS) for distributed sensor networks that takes advantage of spatial and temporal relationships among sensors. The proposed FDS relies on a suitable functional graph representation of the network and a two-layer hierarchical architecture designed to promptly detect and isolate faults. The lower processing layer exploits a novel change detection test (CDT) based on hidden Markov models (HMMs) configured to detect variations in the relationships between couples of sensors. HMMs work in the parameter space of linear time-invariant dynamic systems, approximating, over time, the relationship between two sensors; changes in the approximating model are detected by inspecting the HMM likelihood. Information provided by the CDT layer is then passed to the cognitive one, which, by exploiting the graph representation of the network, aggregates information to discriminate among faults, changes in the environment, and false positives induced by the model bias of the HMMs.