How accurate are percentile measures of less than one percent in a normal distribution? How confident can you be that a test is reliable (and measures what it is supposed to) if the number of sample units (in a random sample) above 1% or below 1% is small? There might be many sample units around the mean or median but as you approach the tails of the distribution <1% there are fewer and fewer sample units. So how large should a sample size be to accurately measure phenomena that occurs less than 1% of the time? Isn't the test supposed to measure the mean of the observed phenomena and we can't be confident that extreme values measure what they supposed to?
For example, you might want to measure rainfall patterns. You have a large amount of data that surrounds the mean. But you have very few sample units of extreme weather that occurs less than 1% of the time. This type of extreme weather might not happen for many years until it does. So how would you talk about how reliably you can predict this type of extreme weather? Say you wanted to give a 99.5% or 99.99% probability of a certain extreme weather event occurring. I don't think you could have much precision or reliably make predictions at these extreme percentiles (the only way would be to increase the sample size so that you can record a high number of extreme weather events). Unlike you can have close to the mean and median because there is much more data available and you can have a high confidence that the what you are measuring is actually empirically accurate.
I'm assuming that rainfall follows a normal distribution. But I'm not sure if it does. You could take human height or whatever other data sets are normally distributed (approximately).
 A: Your intuition is right that you need larger sample sizes to estimate the tails of distributions. Tolerance intervals are a common way to estimate uncertainty on quantiles of a distribution. It is simply a confidence interval on a quantile. Sample sizes can be computed by specifying the amount of precision you want in this confidence interval. Such calculations can be done in many software packages. For example, the R package 'tolerance' has some capability for this. 
Providing an exact answer to your question of sample size is precarious. Yes, it can be done if you assume a normal distribution, but estimating tails is notoriously difficult compared to estimating means and medians...and getting the distributional assumption even slightly wrong can change the results significantly. The catch-22 is you have to check the distributional fit in the tails - precisely where you have little data to do the checking. 
On another note - your example of rainfall is a common application in 'extreme value theory' and you should consider looking there to help in your particular application.
