Humans and macaque monkeys adjust their response time adaptively in stop-signal (countermanding) tasks, responding slower after stop-signal trials than after control trials with no stop signal. We investigated the neural mechanism underlying this adaptive response time adjustment in macaque monkeys performing a saccade countermanding task. Earlier research showed that movements are initiated when the random accumulation of presaccadic movement-related activity reaches a fixed threshold. We found that a systematic delay in response time after stop-signal trials was accomplished not through a change of threshold, baseline, or accumulation rate, but instead through a change in the time when activity first began to accumulate. The neurons underlying movement initiation have been identified with stochastic accumulator models of response time performance. Therefore, this new result provides surprising new insights into the neural instantiation of stochastic accumulator models and the mechanisms through which executive control can be exerted.