No, but when you get into the nitty gritty of most things sometimes being influenced by extremely rare things, and also that the convergence rate of the central limit theorem is not universal at all, then much of the utility (and apparent universality) of the CLT starts to evaporate.

In practice when modeling you are almost always better not assuming normality, and you want to test models that allow the possibility of heavy tails. The CLT is an approximation, and modern robust methods or Bayesian methods that don't assume Gaussian priors are almost always better models. But this of course brings into question the very universality of the CLT (i.e. it is natural in math, but not really in nature).

Heavy tails are everywhere. Normal distributions have absurdly light tails. Levy alpha stable distributions have power law tails. Power law tails are everywhere.

Some things with heavy tails:

  token occurrences
  comment thread upvotes
  startup IPOs
  social follower counts
  network latency
  github stars
  git diffs
  power station size
  weather events