Robustness, a relative insensitivity to perturbations, is a key characteristic of living cells. However, the specific structural characteristics that are responsible for robust performance are not clear, even in genetic circuits of moderate complexity. Formal sensitivity analysis allows the investigation of robustness and fragility properties of mathematical models representing regulatory networks, but it yields only local properties with respect to a particular choice of parameter values. Here, we show that by systematically investigating the parameter space, more global properties linked to network structure can be derived. Our analysis focuses on the genetic oscillator responsible for generating circadian rhythms in Drosophila as a prototypic dynamical cellular system. Analysis of two mathematical models of moderate complexity shows that the tradeoff between robustness and fragility is largely determined by the regulatory structure. Rank-ordered sensitivities, for instance, allow the correct identification of protein phosphorylation as an influential process determining the oscillator's period. Furthermore, sensitivity analysis confirms the theoretical insight that hierarchical control might be important for achieving robustness. The complex feedback structures encountered in vivo, however, do not seem to enhance robustness per se but confer robust precision and adjustability of the clock while avoiding catastrophic failure.