When making choices, collecting more information is beneficial but comes at the cost of sacrificing time that could be allocated to making other potentially rewarding decisions. To investigate how the brain balances these costs and benefits, we conducted a series of novel experiments in humans and simulated various computational models. Under six levels of time pressure, subjects made decisions either by integrating sensory information over time or by dynamically combining sensory and reward information over time. We found that during sensory integration, time pressure reduced performance as the deadline approached, and choice was more strongly influenced by the most recent sensory evidence. By fitting performance and reaction time with various models we found that our experimental results are more compatible with leaky integration of sensory information with an urgency signal or a decision process based on stochastic transitions between discrete states modulated by an urgency signal. When combining sensory and reward information, subjects spent less time on integration than optimally prescribed when reward decreased slowly over time, and the most recent evidence did not have the maximal influence on choice. The suboptimal pattern of reaction time was partially mitigated in an equivalent control experiment in which sensory integration over time was not required, indicating that the suboptimal response time was influenced by the perception of imperfect sensory integration. Meanwhile, during combination of sensory and reward information, performance did not drop as the deadline approached, and response time was not different between correct and incorrect trials. These results indicate a decision process different from what is involved in the integration of sensory information over time. Together, our results not only reveal limitations in sensory integration over time but also illustrate how these limitations influence dynamic combination of sensory and reward information.