Solution:

The sum of the first $n$ numbers is $$\dfrac{n(n + 1)}{2}.$$ We need this to not divide $n!.$

Note that if $n + 1$ is composite, then it can be broken down into factors that divide $n!.$

This means that we need $n + 1$ to be prime. The largest three-digit prime is $997,$ so the largest $n$ value is $$997 - 1 = 996.$$

Thus, B is the correct answer.