Previous research and methodological advice has focused on the importance of accounting for measurement error in psychological data. That perspective assumes that psychological variables conform to a common factor model. We explore what happens when data that are not generated from a common factor model are nonetheless modeled as reflecting a common factor. Through a series of hypothetical examples and an empirical reanalysis, we show that when a common factor model is misused, structural parameter estimates that indicate the relations among psychological constructs can be severely biased. Moreover, this bias can arise even when model fit is perfect. In some situations, composite models perform better than common factor models. These demonstrations point to a need for models to be justified on substantive, theoretical bases in addition to statistical ones. (PsycINFO Database Record (c) 2020 APA, all rights reserved).