The t-tests are interpretable when the mathematical assumptions are met as well as when the results and data in the context of the experiment or inference make sense as well.
Statistical tests are based on assumptions and if you put garbage in, you'll get garbage out as well.
The t-test has a specific distribution (Student-t distribution) and this distribution depends on s^2 having roughly a chi-square distribution and the sample mean having roughly a normal distribution where s^2 is independent from the sample mean. If these are not met, you can't use a t-distribution.
The Central Limit Theorem can give some sort of guarantee for the distribution of the sample mean (in fact the CLT is the reason all the frequentist statistics actually work) and in some respects it also does the same for the variance as well (recall that the sum of squared normals is what a chi-square is) but this depends on the distribution since really skewed distributions will require a lot more samples to get close to these assumptions than less skewed ones.