We give a sharp lower bound for the Hilbert function in degree d of artinian quotients k[x1, … , xn] / I failing the Strong Lefschetz property, where I is a monomial ideal generated in degree d≥ 2. We also provide sharp lower bounds for other classes of ideals, and connect our result to the classification of the Hilbert functions forcing the Strong Lefschetz property by Zanello and Zylinski.