Five-Percent Days the Bell Curve Forbids

An equity index has an annualised volatility of $16\%$, and there are about $252$ trading days in a year. Model daily log returns as independent and normally distributed with that volatility.

(a) Under this model, how often should the index move more than $3\%$ in a single day? More than $5\%$? (b) The real index has had well over a dozen daily moves beyond $5\%$ in the past century, several of them within the same few weeks. What is wrong with the model, and what is the smallest change that repairs it?