Summary: Let \(d\) be an integer and let \(G\) be a 2-group of finite exponent whose \(d\)-generator subgroups are nilpotent of class at most a fixed integer \(c\). Free groups of infinite rank in the variety defined by \(X^ 4=1\) show that if \(d>2\) and \(c=3d-2\), \(G\) is not necessarily solvable. On the other hand, we prove that \(G\) is solvable if \(d>0\) and \(c=3d-3\).