In this paper, an algorithm with polynomial timecomplexity is presented for testing failure diagnosability in (untimed) discrete-event systems in a state-based framework. Furthermore, an algorithm for testing failure diagnosability in timed discrete-event systems is provided. The test for timed discrete-event systems, in particular, first gathers and complies the information about the timing of events (represented in the timed transition graph of the timed system) in the transition-time function of a reduced model, and then uses this model to verify diagnosability. Sufficient conditions are obtained under which the transition-time sets can be represented as the union of a bounded number of intervals, and the test will have polynomial complexity. This new test, as shown using examples, may significantly reduce the computations of testing diagnosability, compared with other polynomial diagnosability tests (for untimed systems) adapted for timed systems.