Actually we don't take a sample from a normal distribution. We take it from whatever distribution the population we sample from happens to be. We assume, often, that the distribution is normal, because that is mathematically convenient. But that does not make the sample or population distribution normal. And if the real population is not normal, and that is common, the statistics we use that assume a normal distortion will be wrong to some extent. This is particularly obvious with outlier analysis which commonly assumes normal distribution - when the distribution is not normal the outliers you generate won't really be outliers from the actual population you sampled from.

With large samples normality rarely if ever matters because statistics are asymptotically correct even with non-normality (this is tied in part to the central limit theorem).