The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call minimal non-N . To facilitate this we investigate solvable Lie algebras of nilpotent length k, and of nilpotent length ≤ k, and extreme Lie algebras, which have the property that their nilpotent length is equal to the number of conjugacy classes of maximal subalgebras. We characterise the minimal non-N Lie algebras in which every nilpotent subalgebra is abelian, and those of solvability index ≤ 3.