Compared to traditional confirmatory factor analysis (CFA), exploratory structural equation modeling (ESEM) has been shown to result in less structural parameter bias when cross-loadings (CLs) are present. However, when model fit is reasonable for CFA (over ESEM), CFA should be preferred on the basis of parsimony. Using simulations, the current study examined the sensitivity of the CFI, RMSEA, and RMSEAD in correctly adjudicating model fit between ESEM and CFA. Results showed that 1) the magnitude of structural bias was moderated by the sign of the CL, 2) constraining non-zero CL to zero resulted in incorrectly specified CFAs demonstrating good stand-alone fit but were often rejected when compared with ESEM, and 3) CFAs with negligible factor correlation bias <|0.10| often failed the model equivalence test while those with non-ignorable bias >|0.30| passed. This disconnect is shown to be linked to CL conditions and calls into question historically held beliefs about what constitutes an "ignorable" CL value.