Let T(bjk) be a finite collection of infinite Toeplitz band matrices, let Tn(bjk) denote their (n+1) x (n+1) truncations, and put \[ A_n=\sum_j\prod_k T_n(b_{jk}),\quad M_p=\lim_{n\to\infty}\|A_n\|_p, \] where $\|\cdot\|_p$ stands for the operator norm associated with the $l^p$-norm $(1\le p<\infty)$ on Cn+1. We establish tight two-sided estimates for the difference Mp-||An||p. It is well known that if T(b) is a Hermitian Toeplitz band matrix and An=Tn(b), then M2-||An||2 goes to zero with polynomial speed. We show that such a slow convergence rate is, in a sense, an exceptional case, and we prove that in the generic case Mp - ||An||p approaches zero with exponential speed. Our results yield good error estimates when computing the norms of certain infinite matrices via their large truncations and, conversely, when determining the norms of certain large matrices by having recourse to known norms of infinite matrices.