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System-level diagnosis is a crucial subject for maintaining the reliability of multiprocessor interconnected systems. Consider a system composed of N independent processors, each of which tests a subset of the others. Under the PMC diagnosis model, Dahbura and Masson proposed an O(N2.5) algorithm to identify the set of faulty processors in a t-diagnosable system, in which at most t processors are permanently faulty. In this paper, we establish some sufficient conditions so that a t-regular system can be conditionally (2t-1)-diagnosable, provided every fault-free processor has at least one fault-free neighbor. Because any t-regular system is no more than t-diagnosable, the approached diagnostic capability is nearly double the classical one-step diagnosability. Furthermore, a correct and complete method is given which exploits these conditions and the presented branch-of-tree architecture to determine the fault status of any single processor. The proposed method has time complexity O(t2), and thus can diagnose the whole system in time O(t2 N). In short, not only could the diagnostic capability be proved theoretically, but also it is feasible from an algorithmic perspective.