Let $\mathcal{C}\subseteq \mathbb{F}_{q^m}^n$ be an $\mathbb{F}_{q^m}$-linear non-degenerate rank metric code with dimension $k$. In this paper we investigate the problem of determining the number $M(\mathcal{C})$ of codewords in $\mathcal{C}$ with maximum weight, that is $\min\{m,n\}$, and to characterize those with the maximum and the minimum values of $M(\mathcal{C})$.