Suppose, for concreteness, that I am trying to estimate the mean of a population using a random sample of size $N$.
Many elementary books discuss forming a confidence interval for the population mean by using the central limit theorem to argue that the sample mean is approximately normally distributed.
However, the central limit theorem is a theorem about the limit as $N \to \infty$. But if $N$ were really large then the width of the confidence interval would be really small and giving just a point estimate would be good enough.
So it seems that we are implicitly assuming that there is a range of $N$ for which the CLT is not too bad an assumption but for which $N$ is not so large as to shrink the confidence interval to practically a point.
My question is: is there any basis for making this implicit assumption? Is there any way to judge what this range of useful $N$ is for a particular application?