Let $p$ a prime number. Let $G$ be a $p$-group finitely generated, say, $G = \left<x_1, x_2, \ldots, x_m \right>$. Denote by $\gamma_i(G)$ the $ith$ term of the lower central series of $G$. Then, is true that $\gamma_i(G)$ is generated by a number of elements that depends only $m$ and $i$?
I think the answer is no. Let $G$ be a wreath product of a cyclic group of order $p$ with a cyclic group of order $p^k$. Then the derived group $\gamma_2(G)$ is elementary abelian of rank $p^k-1$. There is of course a bound as a function of $m$, $i$ and the nilpotency class $c$.