Disjoint paths of length at most L and the number of nodes to remove to vanish this property - Inequality
Hint: Suppose that $P_1,\ldots,P_t$ are internally disjoint $(s,t)$-paths of length at most $L$. Show that you have to remove at least one vertex from each path to make the distance from $s$ to $t$ larger than $L$.

When $L = |G|$, the classical Menger's theorem states that equality holds.