Is there a notion of reliability for percentiles?

Say I run an experiment measuring some parameter A on a sample group of a population where A is a non-negative real number. Is there some way of knowing if my percentile results are reasonable? Ideally, I'd have confidence intervals around my percentiles to know their ranges in the population as a whole instead of just in the sample group.

TL;DR want confidence intervals for percentiles

If your estimate of the of the $p$ quantile of the population is observation $np$ of the sorted sample (where $n$ is the number of observations) then a reasonable approximation of say the 95% confidence interval is from sorted observation $np - 1.96 \sqrt{np(1-p)}$ to sorted observation $np + 1.96 \sqrt{np(1-p)}$.
If $n$ is small then there are better approximations, both for the quantile itself which will typically involve interpolation between observations and for the confidence interval which can use the binomial distribution rather than the normal approximation to it.