The central limit theorems

There are multiple versions of the central limit theorem. One common version says that, as $n\to \infty$, the mean of means looks like a Gaussian distribution. People in industry sometimes ignore the "goes to infinity" part, and assume that if $n$ is large, that's good enough. But is that true? When is it safe to assume that your means have converged? I heard of a case recently where the mean of means was assumed to have converged, hadn't, and the application it was a part of suffered accuracy problems for months without anyone knowing. This sequence will explore: under what conditions does the central limit theorem "fail" like this?