The availability of large-scale population genomic sequence data has resulted in an explosion in efforts to infer the demographic histories of natural populations across a broad range of organisms. As demographic events alter coalescent genealogies they leave detectable signatures in patterns of genetic variation within and between populations. Accordingly, a variety of approaches have been designed to leverage population genetic data to uncover the footprints of demographic change in the genome. The vast majority of these methods make the simplifying assumption that the measures of genetic variation used as their input are unaffected by natural selection. However, natural selection can dramatically skew patterns of variation not only at selected sites, but at linked, neutral loci as well. Here we assess the impact of recent positive selection on demographic inference by characterizing the performance of three popular methods through extensive simulation of datasets with varying numbers of linked selective sweeps. In particular, we examined three different demographic models relevant to a number of species, finding that positive selection can bias parameter estimates of each of these models--often severely. Moreover, we find that selection can lead to incorrect inferences of population size changes when none have occurred. We argue that the amount of recent positive selection required to skew inferences may often be acting in natural populations. These results suggest that demographic studies conducted in many species to date may have exaggerated the extent and frequency of population size changes.