Q:

Example of central limit theorem fail due to dependence (for tuition)

I want to get better understanding why the central limit theorem can fail. I have seen examples that the theorem fails when the variables are not identically distributed, and when the variance is not finite. 
Can you suggest an example where CLT fails due to dependence? It is intended for tuition, so it is good if the example does not seem too contrived, and perhaps uses one of the common continuous probability distributions.

A:

Let $P(X=1)=P(X=-1) = 1/2$ and let the summands $X_n$ all equal $X$. They are obviously identically distributed, but obviously not independent.  Their sum $S_n=X_1+\cdots+X_n$ (which is $nX$) does not have a gaussian limit.  For another example, let $X_n = (-1)^n X$.  Again, identically distributed but not independent. Their sum (in this case $S_n=0$ if $n$ is even, $S_n=-X$ if $n$ is odd) doesn't have a gaussian limit in a different way, with a different scaling.
But maybe these are too contrived?  Maybe, but I think any counterexample will be a bit like these.  The CLT is about cancellation of sums of random variables, and these examples show the extremes of no cancellation and maximal cancellation.  Any counterexample will partake of some of these behaviors.  

