Summary: We study the minimum length-bounded cut problem where the task is to find a set of edges of a graph such that after removal of this set, the shortest path between two prescribed vertices is at least \(L + 1\) long. We show the problem can be computed in \(\mathsf{FPT}\) time with respect to \(L\) and the tree-width of the input graph \(G\) as parameters and with linear dependence of \(|V(G)|\) (i.e., in time \(f(L, \operatorname{tw}(G))|V(G)|\) for a computable function \(f\)). We derive an \(\mathsf{FPT}\) algorithm for a more general multi-commodity length-bounded cut problem when additionally parameterized by the number of terminals. For the former problem we show a \(\mathsf{W}[1]\)-hardness result when the parameterization is done by the path-width only (instead of the tree-width) and that this problem does not admit polynomial kernel when parameterized by path-width and \(L\). We also derive an \(\mathsf{FPT}\) algorithm for the minimum length-bounded cut problem when parameterized by the tree-depth, thus showing an interesting behavior for this problem and parameters tree-depth and path-width.