The assumption is reasonable for a lot of cases because of the central limit theorem. In particular, it applies if the variable of interest is the sum of a large number of independent random variables. For example, the number of heads in N coin flips is the sum of the number of heads in each of N independent experiments each consisting of a single flip.

Of course, there are also a lot of important cases where that N isn't quite large enough to justify the normal distribution, or where the quantity of interest is not a sum of a large number of small effects, so the CLT doesn't apply at all.