Regarding the nilpotency class of finite p-groups

Question

I'm aware that for a $p$-group $G$ of order $p^{n}$, say, that $G$ must have a nilpotency class between 1 and $n - 1$ for $n\geq 2$. My question is why can $G$ not have nilpotency class $n$? Take for example a group of order $p^{3}$. Why is it not possible to make a lower central series with orders $p^{3} \rightarrow p^{2} \rightarrow p \rightarrow 1$? I know all the 5 possible groups in this case have nilpotency class 1 or 2. But since there's theoretically 'room' for there to be a chain of length 4, how do we know this is never the case in general?