This paper discusses the issue of multiple testing and overall Type I error rates in contexts other than multiple comparisons of means. It demonstrates, using a 5 x 5 correlation matrix, the application of 5 recently developed modified Bonferroni procedures developed by the following authors: (1) Y. Hochberg (1988); (2) B. S. Holland and M. D. Copenhaver (1987); (3) S. Holm (1979); (4) G. Hommel (1988); and (5) D. M. Rom (1990). The original Bonferroni procedure is compared with these modified procedures, using 50 correlation matrices reported in educational and psychological journals. A total of 2,587 p-values were obtained. Using an overall alpha per matrix of 0.05, 0.10, and 0.20, the number of rejections were counted. For a relatively large number of matrices, the modified Bonferroni procedures rejected the same number of hypotheses as did the original Bonferroni procedure. However, whenever there were differences, all Bonferroni procedures rejected more hypotheses than did the original procedure. Among the four modified procedures, the Holm procedure had the lowest and the Hommel procedure had the highest number of rejections. The larger the Type I error rate, the greater was the difference among the methods in the number of rejections. The differences in the number of rejections may depend on the distribution of a set of p-values. Thirteen tables contain the analysis results, and there is a 39-item list of references. (Author/SLD)