Let \({\mathcal B}_{n,h,k}\) be the linear space of \(n\times n\) Toeplitz matrices \(A=(a_{i,j})\) such that \(a_{i,j}=0\) if i-j\(\geq k\) or j-i\(\geq h\). Any matrix belonging to \({\mathcal B}_{n,h,k}\) is called a band Toeplitz matrix. Let \({\mathcal L}_{n,k}\) be the subspace of \({\mathcal B}_{n,k,k}\) made up by symmetric matrices. In the main section, the authors determine the tensor rank and the border rank of the classes \({\mathcal B}_{n,h,k}\) and \({\mathcal L}_{n,k}\). The authors show that \(rank_ F({\mathcal B}_{n,h,k})=n+h-1\), if the field F contains a primitive \((n-1+h)^{th}\) proof of unity; [border \(rank]_ F({\mathcal B}_{n,h,k})=n+k-1\) and \(rank_ R({\mathcal L}_{n,k})=[border\) \(rank]_ R({\mathcal L}_{n,k})=n+2[(k-1)/2]\), if R is the real field and \(k\leq n/2\).