Stable and robust oscillations in the concentration of adenosine 3, 5-cyclic monophosphate (cAMP) are observed during the aggregation phase of starvation-induced development in Dictyostelium discoideum. In this paper we use mathematical modelling together with ideas from robust control theory to identify two factors which appear to make crucial contributions to ensuring the robustness of these oscillations. Firstly, we show that stochastic fluctuations in the molecular interactions play an important role in preserving stable oscillations in the face of variations in the kinetics of the intracellular network. Secondly, we show that synchronisation of the aggregating cells through the diffusion of extracellular cAMP is a key factor in ensuring robustness of the oscillatory waves of cAMP observed in Dictyostelium cell cultures to cell-to-cell variations. A striking and quite general implication of the results is that the robustness analysis of models of oscillating biomolecular networks (circadian clocks, Ca(2+) oscillations, etc.) can only be done reliably by using stochastic simulations, even in the case where molecular concentrations are very high.