Do you ever test the pre-asymptotic behaviour of a variable before performing a comparison of means t-test? I understand that it is not the normality of a random variable that matters in a t-test, but rather the fact that the distribution of the mean follows a normal distribution for large samples. However, is it sometimes useful to test how large a sample must be before the assumption of a normal distribution is warranted? For example, if the number of samples required were very large (much larger than n=30), would assuming a normal distribution be inappropriate? If so, how would you go about checking this?
The inspiration for this question comes from reading about Nassim Nicholas Taleb's Kappa metric.

When you are using a statistical test that relies on the central limit theorem (as the T-test does), the test is only valid if the underlying distribution has a finite variance.  If the underlying distribution of the data has sufficiently heavy tails that it does not have a finite variance then the central limit theorem does not apply and the test will not work properly.  Thus, the most important preliminary issue for applying a T-test is whether or not the data come from a distribution where the rate of decay in the tails is sufficiently fast to give finite variance; anything faster than cubic decay is sufficient to give a finite variance.
To examine the rate of decay in the tails of a distribution, using a sample from that distribution, the usual diagnostic tool is a tail plot.  You can find a related post here discussing the construction of a tail plot and the examination of the plot to see if the distribution has a finite variance.  An example of tail plots for the two tails of a dataset is shown below.

