Many decisions benefit from the accumulation of evidence obtained sequentially over time. In such circumstances, the decision-maker must balance speed against accuracy, while considering the cost associated with the passage of time. A neural mechanism to achieve this balance is to accumulate evidence and to terminate the deliberation when enough evidence has accrued. To accommodate time costs, it has been hypothesized that the criterion to terminate a decision may become lax as a function of time. Here we tested this hypothesis by manipulating the cost of time in a perceptual choice-reaction time (RT) task. Human participants (both sexes) discriminated the direction of motion in a dynamic random-dot display, which varied in difficulty across trials. Unbeknownst to the participants, halfway through the experiment, we increased the time pressure by canceling a small fraction of trials, mimicking a broken fixation, if they had not made a decision by a provisional deadline. This subtle manipulation led participants to make faster but less accurate decisions. Choice and RT were well explained by a bounded evidence-accumulation process. We developed a novel computational method to estimate the time-dependent changes in the stopping bounds directly from the participants' RT and choice data. Our analysis revealed that the bounds decline as a function of time, and that this decline is steeper following the time-cost manipulation. The time-varying decision bounds approximate an optimal stopping policy, although the specific bound shape is idiosyncratic across individuals.