The CLT is why you can put a confidence interval on a mean return without knowing the shape of individual trade outcomes. Sum enough independent draws and the sum's distribution smooths out.
Markets break two of its conditions. Returns are not independent, because volatility clusters, and their tails can be heavy enough that the variance is poorly estimated. Convergence then takes far longer than the textbook suggests, and the normal approximation understates how often extreme averages occur.
Practical reading: trust the CLT for the mean of a few hundred well-behaved trade results, distrust it for anything involving the tail, such as max-drawdown or the worst month. For those, simulate rather than assume.
Related: standard-error, fat-tails, law-of-large-numbers, monte-carlo-simulation