Let $\mathcal{B}_{n,h,k} $ be the class of $n \times n$ band Toeplitz matrices $A = (a_{i,j} )$ such that $a_{i,j} = 0$ if $i - j \geqq k$ or $j - i \geqq h$, $k \leqq h$ and $\mathcal{S}_{n,k} $ the subclass of $\mathcal{B}_{n,k,k} $ made up by symmetric matrices. We show that ${\operatorname{rk}}_{\bf F} (\mathcal{B}_{n,h,k} ) = n + h - 1$ if the field ${\bf F}$ contains a primitive $(n - 1 + h)$th root of unity, ${\operatorname{brk}}_{\bf F} (\mathcal{B}_{n,h,k} ) = n + k - 1,{\operatorname{rk}}_{\bf R} (\mathcal{S}_{n,k} ) = {\operatorname{brk}}_{\bf R} (\mathcal{S}_{n,k} ) = n + 2\lfloor {{(k - 1)} / 2} \rfloor $ if ${\bf R}$ is the real field and $k \leqq {n / 2}$.