We determine the conditions in which sustained oscillations develop in a model for a bicyclic enzyme cascade regulated by negative feedback. The model, based on a cascade of two phosphorylation-dephosphorylation cycles, was previously proposed (Goldbeter, A. Proc. Natl. Acad. Sci. U.S.A. 1991, 88, 9107) as a minimal cascade model for the mitotic oscillator driving the early cell division cycles in amphibian embryos. We analyze the role of thresholds in the mechanism of oscillatory behavior by constructing stability diagrams as a function of the main parameters of the model. The thresholds arise from the phenomenon of zero-order ultrasensitivity naturally associated with the kinetics of phosphorylation-dephosphorylation cycles (Goldbeter, A.; Koshland, D. E., Jr. Proc. Natl. Acad. Sci. U.S.A. 1981, 78, 6840). The analysis shows that if the existence of a threshold in each of the two cycles markedly favors the periodic operation of the cascade, a single threshold suffices for sustained oscillatory behavior. Oscillations may even arise in the absence of any threshold in a small region of parameter space, but their amplitude is very much reduced. The model provides an example of biochemical oscillator based on negative feedback in which nonlinear amplification, instead of being due to allosteric cooperativity, results from the ultrasensitivity that arises from the kinetics of phosphorylation-dephosphorylation cycles.