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Previous research and methodological advice has focussed on the importance of accounting for measurement error in psychological data. That perspective assumes that psychological variables conform to a common factor model, such that they consist of construct variance plus error. In this paper, we ask what happens when a set of items that are not generated from a common factor construct model are nonetheless modeled as reflecting a common factor. Through a series of hypothetical examples and a real-data re-analysis, we show that (1) common factor models tend to produce extremely biased and highly variable structural parameter estimates when the population model is not a common factor; (2) model fit is a poor indicator of the degree of bias; and (3) composite models are sometimes more reliable than common factor models under alternative measurement structures, though they also lead to unacceptably bad solutions in some cases.