This paper is devoted to studying the asymptotical features of the standard statistical test (sometimes called the t-test for the correlation coefficient) for verifying the hypothesis about the significance of the coefficient of Pearson correlation between random variables x and y. Despite the fact that this test has been substantiated only under the assumption of a Gaussian character for the joint distribution of x and y, it is very widely used and incorporated in most statistical packages. However, the assumption about a Gaussian character of distributions usually fails in practice, so a problem exists with describing the applicability region of the t-test at great sample sizes. It has been proven in this work that this test is asymptotically exact for independent x and y when certain additional conditions are met, whereas a simple lack of correlation may be insufficient for such a feature. In addition, an asymptotically exact and consistent test has been constructed in the absence of independence. Computational experiments argue for its applicability in practice. Moreover, these results have been extended to the partial correlation coefficient after corresponding modifications.