The endogenous circadian clock synchronizes with environmental time by appropriately resetting its phase in response to external cues. Of note, some resetting stimuli induce attenuated oscillations of clock output, which has been observed at the population-level in several organisms and in studies of individual humans. To investigate what is happening in individual cellular clocks, we studied the unicellular cyanobacterium S. elongatus. By measuring its phase-resetting responses to temperature changes, we found that population-level arrhythmicity occurs when certain perturbations cause stochastic phases of oscillations in individual cells. Combining modeling with experiments, we related stochastic phasing to the dynamical structure of the cyanobacterial clock as an oscillator and explored the physiological relevance of the oscillator structure for accurately timed rhythmicity in changing environmental conditions. Our findings and approach can be applied to other biological oscillators.