Stimulus from the environment that guides behavior and informs decisions is encoded in the firing rates of neural populations. Neurons in the populations, however, do not spike independently: spike events are correlated from cell to cell. To what degree does this apparent redundancy have an impact on the accuracy with which decisions can be made and the computations required to optimally decide? We explore these questions for two illustrative models of correlation among cells. Each model is statistically identical at the level of pairwise correlations but differs in higher-order statistics that describe the simultaneous activity of larger cell groups. We find that the presence of correlations can diminish the performance attained by an ideal decision maker to either a small or large extent, depending on the nature of the higher-order correlations. Moreover, although this optimal performance can in some cases be obtained using the standard integration-to-bound operation, in others it requires a nonlinear computation on incoming spikes. Overall, we conclude that a given level of pairwise correlations, even when restricted to identical neural populations, may not always indicate redundancies that diminish decision-making performance.