In this work we investigate nilpotent groups G in which all proper subgroups (or all subgroups of infinite index) have class smaller than the class of G. Our main results are obtained by considering analagous questions for Lie algebras and using the Lazard correspondence and the Mal'cev correspondence. Among other things, for each n≥3, we prove the existence of nilpotent groups of class 2n in which every proper subgroup (or subgroup of infinite index) has class at most n.