Summary: Let \(G\) be a nilpotent group with cyclic commutator subgroup of order \(p^n\) and let \(F\) be a field of characteristic \(p\). It is shown here that the Lie derived length of the group algebra \(FG\) is at most \(\lceil\log_2(p^n+1)\rceil\). Furthermore, this bound is achieved if and only if one of the following conditions is satisfied: (i) \(p\) is odd; (ii) \(p=2\) and \(n\leq 2\); (iii) \(p=2\), \(n\geq 3\) and the nilpotency class of \(G\) is at most \(n\).