An improvement of a bound of B. Yankosky [J. Lie Theory 13, No. 1, 1–6 (2003; Zbl 1010.17005)] is presented in this paper, thanks to a restriction which has been recently obtained by the authors on the Schur multiplier M(L) of a finite dimensional nilpotent Lie algebra L. It is also described the structure of all nilpotent Lie algebras such that the bound is attained. An important role is played by the presence of a derived subalgebra of maximal dimension. This allows precision on the size of M(L). Among other results, applications to the non-abelian tensor square L⊗L are illustrated.