Speed-accuracy tradeoff (SAT) is an adaptive process balancing urgency and caution when making decisions. Computational cognitive theories, known as evidence accumulation models, have explained SATs via a manipulation of the amount of evidence necessary to trigger response selection. New light has been shed on these processes by single-cell recordings from monkeys who were adjusting their SAT settings. Those data have been interpreted as inconsistent with existing evidence accumulation theories, prompting the addition of new mechanisms to the models. We show that this interpretation was wrong, by demonstrating that the neural spiking data, and the behavioural data are consistent with existing evidence accumulation theories, without positing additional mechanisms. Our approach succeeds by using the neural data to provide constraints on the cognitive model. Open questions remain about the locus of the link between certain elements of the cognitive models and the neurophysiology, and about the relationship between activity in cortical neurons identified with decision-making vs. activity in downstream areas more closely linked with motor effectors. 