It is numerically well known that moment-based tests for Gaussianity and estimators become increasingly unreliable at higher moment orders; however, this phenomenon has lacked rigorous mathematical justification. In this work, we establish quantitative bounds for Hermite-based moment tests, with matching exponential upper and lower bounds. Our results show that, even under ideal conditions with i.i.d. standard normal data, the sample size must grow exponentially with the highest moment order $d$ used in the test. These bounds, derived under both the convex distance and the Kolmogorov-Smirnov distance, are applied to classical procedures, such as the Shenton-Bowman test.