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Fit indices are commonly used in structural equation modeling to assess model-data fit and evaluate the relative fit of competing models. Obviously the assumption of uncorrelated error terms may be violated in some circumstances and this will lead to model misspecifications. The ability to show failure of commonly used fit indices, such as RMSEA and SRMR, at some given cut-off values, to flag model misspecifications has demonstrated the value of this piece of research article. The results of this study show that the commonly used global goodness-of-fit indexes do not provide a sound basis for the assessment of model fit when the assumption of uncorrelated errors is violated. The central idea throughout the article is that a model that is viewed as well-fitting according to global model fit indices can actually be seriously misspecified with regard to correlated errors. This result has led us to ask a question: Are there any alternative approaches for assessing model fit when correlated errors are likely to occur? By the way, further research could be done with more complex factor structures that move beyond simply two-factor structure.