For discrete event systems, we study the problem of predicting failures prior to their occurrence, also referred to as prognosis, in the inference-based decentralized framework where multiple decision-makers interact to make the global prognostic decisions. We characterize the class of systems for which there are no missed detections (all failures can be prognosed prior to their occurrence) and no false alarms (all prognostic decisions are correct) by introducing the notion of N-inference-prognosability, where the parameter represents the maximum ambiguity level of any winning prognostic decision. An algorithm for verifying N-inference-prognosability is presented. We also show that the notion of coprognosability introduced in our prior work is the same as 0-inference-prognosability, and as the parameter N is increased, a larger class of prognosable systems is obtained.