<p><it>Abstract—</it>In the system-level fault diagnosis area, the fundamental problem of characterizing sequentially $<tmath>t</tmath>$-diagnosable systems in the PMC model has remained open for more than two decades. We resolve this problem by providing a complete characterization of such systems. Our solution to the characterization problem leads to the correct identification of optimal sequentially $<tmath>t</tmath>$-diagnosable $<tmath>D_{\delta,k}</tmath>$ systems. Given a set of $<tmath>n</tmath>$ units where $<tmath>n = 2t + 1</tmath>$, an optimal $<tmath>D_{\delta,k}</tmath>$ system can be constructed with just $<tmath>n(\lfloor(t + 2)/3\rfloor)</tmath>$ tests, rather than $<tmath>n(\lfloor t/2\rfloor + 1)</tmath>$ tests—a previously misjudged bound. An efficient algorithm for identifying the set of faulty units in a sequentially $<tmath>t</tmath>$-diagnosable $<tmath>D_{\delta,k}</tmath>$ system is given along the line of the proposed characterization, which is linear with respect to the number of tests in the system.</p><p><it>Index Terms—</it>Consistent fault sets, $<tmath>D_{\delta,k}</tmath>$ systems, fault diagnosis, PMC model, sequentially diagnosable systems, syndrome, test assignment.</p>