A unique challenge in air pollution cohort studies and similar applications in environmental epidemiology is that exposure is not measured directly at subjects' locations. Instead, pollution data from monitoring stations at some distance from the study subjects are used to predict exposures, and these predicted exposures are used to estimate the health effect parameter of interest. It is usually assumed that minimizing the error in predicting the true exposure will improve health effect estimation. We show in a simulation study that this is not always the case. We interpret our results in light of recently developed statistical theory for measurement error, and we discuss implications for the design and analysis of epidemiologic research.