Event-based prospective memory (PM) requires a deferred action to be performed when a target event is encountered in the future. Individuals are often slower to perform a concurrent ongoing task when they have PM task requirements relative to performing the ongoing task in isolation. Theories differ in their detailed interpretations of this PM cost, but all assume that the PM task shares limited-capacity resources with the ongoing task. In what was interpreted as support of this core assumption, diffusion model fits reported by Boywitt and Rummel (2012) and Horn, Bayen, and Smith (2011) indicated that PM demands reduced the rate of accumulation of evidence about ongoing task choices. We revaluate this support by fitting both the diffusion and linear ballistic accumulator (Brown & Heathcote, 2008) models to these same data sets and 2 new data sets better suited to model fitting. There was little effect of PM demands on evidence accumulation rates, but PM demands consistently increased the evidence required for ongoing task response selection (response thresholds). A further analysis of data reported by Lourenço, White, and Maylor (2013) found that participants differentially adjusted their response thresholds to slow responses associated with stimuli potentially containing PM targets. These findings are consistent with a delay theory account of costs, which contends that individuals slow ongoing task responses to allow more time for PM response selection to occur. Our results call for a fundamental reevaluation of current capacity-sharing theories of PM cost that until now have dominated the PM literature.