Does there exist context-free grammar with words of length n^2 or n^3?
For a language $L$, let $N(L) = \{|w| : w \in L\}$, and let $U(L) = \{1^n : n \in N(L)\}$. Parikh's theorem shows that if $L$ is context-free then $U(L)$ is regular. In particular, since $\{1^{n^2} : n \geq 0 \}$ is not regular, no context-free language $L$ satisfies $N(L) = \{n^2 : n \geq 0\}$. In fact, no infinite context-free language $L$ even satisfies $N(L) \subseteq \{n^2 : n \geq 0\}$ (exercise).