In this paper, the lack of robustness of the Student's t statistic for long-tailed underlying distributions such as the Student's t with small degrees of freedom is demonstrated. Classical approaches for deriving the distribution of Student's t from a nonnormal universe (like those advanced by Gayen (I), Tiku (5), Srivastava (2) and also those based on an expansion of the underlying density) depend on the existence of moments up to a certain order, which also determines the accuracy of the approximation. Unfortunately, when the parent is Student's t with small degrees of freedom, these procedures either fail or do not work well. A Monte Carlo method with appropriate swindle is applied to obtain the percentage points of this distribution. These are compared with Gayen's approximation. An empirical formula for these percentiles is also obtained which works well but unfortunately depends on the α-level of the test.