Although many parametric statistical tests are considered to be robust, as recently shown in Methodologist's Corner, it still pays to be circumspect about the assumptions underlying statistical tests. In this paper I show that robustness mainly refers to "[alpha]", the type-I error. If the underlying distribution of data is ignored there can be a major penalty in terms of the "[beta]", the type-II error, representing a large increase in false negative rate or, equivalently, a severe loss of power of the test. (Contains 3 figures.)