Application of Central Limit Theorem

This is a bit of a specialist thing; I'd like to hear the opinion of someone with relevant knowledge.

I've never been much mathematically inclined, and now I am seriously rusty, so I can't find my way out of this problem.

Suppose I am measuring a quantity using an instrument for which the manufacturer declares +/- 10% (or any other value) uncertainty: I take a series of measurements, calculate the standard deviation and realize that, in relative terms, it is less than 10%. Standard error of the mean will be even smaller.

So my question is, which result is the correct one? If the central limit theorem applies, the SEM for the sample should be the correct value.

In any case, back then when I did metrology, relative uncertainty was calculated as the square root of the sum of all contributions, among them SEM for the sample and instrumental uncertainty. That one is a very conservative method.