# Reporting an uncertainty lower than the precision of the apparatus?

When performing $N$ independent measurements following a Gaussian distribution, it is sometimes said to report the mean and the standard deviation of the mean as value and its uncertainty. Now, the instrument has a non zero precision, and the standard deviation of the mean can be made arbitrarily close to 0 granted that the number of measurements can be made arbitrarily large. I had been taught to report the precision of the instrument instead of the standard deviation of the mean when this latter becomes smaller than the former, on the basis that "one cannot report the period of a pendulum with an error smaller than the precision of the chronometer used". I feel this rule is arbitrary and I don't find information about it. I also don't see reports where the uncertainty is smaller than the precision of the apparatus. My question is, what do scientists report as uncertainty when the standard deviation of the mean is smaller than the precision of the instrument? I see no problem reporting a number arbitrarily close to 0 as long as it's clear than I'm reporting the standard deviation of the mean. I'd also have no problem reporting another statistics, e.g. the standard deviation to indicate how spread the data is, but then I'm not reporting the uncertainty of the mean value anymore.

$$x = 1.5 \pm 0.15\,(\text{statistical}) \pm 0.2\,(\text{systematic})$$
$\Delta x = \sqrt{\sigma ^2 + \delta ^2}$
Where $\sigma$ is a standard deviation and $\delta$ is a measurement device precision.