Fault management in Web Services composed by individual services from multiple suppliers currently relies on a local analysis, that does not span across individual services, thus limiting the effectiveness of recovery strategies. We propose to address this limitation of current standards for Web Service composition by employing Model-Based Diagnosis to enhance fault analysis. We propose to add Diagnostic Web Services to the set of Web Services providing the overall service, acting as supervisors of their execution, by identifying anomalies and explaining them in terms of faults to be repaired. This approach poses the basis for the development of specialized recovery and compensation techniques aimed at addressing different problems, which could not be otherwise discriminated.