Sample Size and Central Limit Theorem

I am trying to reproduce results from Zachary R. Smith and Craig S. Wells's Central Limit Theorem and Sample Size.

Therein the authors claim the central limit theorem does not hold for samples of size 30 or so (they go so far as to claim 300). I have tried reproducing this claim, their work is unreproducible. Moreover, their work is visually wrong, you can literally plot the sums of uniformly distributed variables and observe the sum is normally distributed.

For those interested in the sordid details, I have posted it on github.

Why did they get this so wrong? Well, they failed to adequately handle multicomparison problems, and they ignored the issues involved with the Kolmogorov-Smirnov test. The latter was particularly troublesome, as it led them to completely erroneous findings.