This research is concerned with two topics in assessing model fit for categorical data analysis. The first topic involves the application of a limited-information overall test, introduced in the item response theory literature, to Structural Equation Modeling (SEM) of categorical outcome variables. Most popular SEM test statistics assess how well the model reproduces estimated polychoric correlations. In contrast, limited-information test statistics assess how well the underlying categorical data are reproduced. Here, the recently introduced C[subscript 2] statistic of Cai and Monroe (2014) is applied. The second topic concerns how the Root Mean Square Error of Approximation (RMSEA) fit index can be affected by the number of categories in the outcome variable. This relationship creates challenges for interpreting RMSEA. While the two topics initially appear unrelated, they may conveniently be studied in tandem since RMSEA is based on an overall test statistic, such as C[subscript 2]. The results are illustrated with an empirical application to data from a large-scale educational survey. [This paper was published in "Multivariate Behavioral Research" v50 n6 p569-583 2015.]