jmc

The smallest sum that can be obtained from $n$ consecutive positive integers is $1+2+\cdots +n=n(n+1)/2$, so we want to find the largest $n$ such that $n(n+1)/2<400$.

Testing, we find that when $n=27$, $n(n+1)/2=27\cdot 28/2=378$, and when $n=28$, $n(n+1)/2=28\cdot 29/2=406$, so the largest such $n$ is $n=\boxed{27}$.