In recent years, classical discrete event fault diagnosis techniques have been extended to Petri Net system models under partial order semantics. In a recent paper, we showed how to take further advantage of the partial order representation of concurrent processes, by decomposing the unfolding into 'facets', formed by subnets whose events either all occur eventually, or none of them occurs. A notion of q(ualitative)-diagnosability was proposed based on this decomposition. The present paper corrects the definition of q-diagnosability and develops its properties. Sufficient and necessary criteria, on the transition labeling, for q-diagnosability are shown; for their verification, and diagnosis itself, compact data structures are sufficient.