In recent years, confirmatory factor analytic (CFA) techniques have become the most common method of testing for measurement equivalence/invariance (ME/I). However, no study has simulated data with known differences to determine how well these CFA techniques perform. This study utilizes data with a variety of known simulated differences in factor loadings to determine how well traditional tests of ME/I can detect these specific simulated differences. Results show that traditional CFA tests of ME/I perform well under ideal situations but that large sample sizes, a sufficient number of manifest indicators, and at least moderate communalities are crucial for assurance that ME/I conditions exist.