[gambling8nt]

Chi-squared distributions are well approximated by normal distributions close to the mean.

The point is not that arbitrary statistics will necessarily always be perfectly behaved (or even well behaved) on sampling data--it's that to make reasonably accurate predictions of system behavior, under certain practical conditions, these statistics are well-behaved, and an inexperienced statistician (as most people are) is less likely to make a gross error.

[DaniFong]

Practical conditions not including routing networks and the stock market, you may wish to add...

[gambling8nt]

Real life situations have finite cutoffs in behavior that remove many pathological problems with certain statistical models.