The ability to trade accuracy for speed is fundamental to human decision making. The speed–accuracy trade-off (SAT) effect has received decades of study, and is well understood in relatively simple decisions: collecting more evidence before making a decision allows one to be more accurate but also slower. The SAT in more complex paradigms has been given less attention, largely due to limits in the models and statistics that can be applied to such tasks. Here, we have conducted the first analysis of the SAT in multiple signal processing, using recently developed technologies for measuring capacity that take into account both response time and choice probability. We show that the primary influence of caution in our redundant-target experiments is on the threshold amount of evidence required to trigger a response. However, in a departure from the usual SAT effect, we found that participants strategically ignored redundant information when they were forced to respond quickly, but only when the additional stimulus was reliably redundant. Interestingly, because the capacity of the system was severely limited on redundant-target trials, ignoring additional targets meant that processing was more efficient when making fast decisions than when making slow and accurate decisions, where participants’ limited resources had to be divided between the 2 stimuli.