A chain S_0 < S_1 < ... < S_n = L is a maximal chain if each S_i is a maximal subalgebra of S_{i+1}. The subalgebra S_0 in such a series is called an n-maximal subalgebra. There are many interesting results concerning the question of what certain intrinsic properties of the maximal subalgebras of a Lie algebra L imply about the structure of L itself. Here we consider whether similar results can be obtained by imposing conditions on the n-maximal subalgebras of L, where n>1.