We show that if $r^n (hA),nh = t$, is an A-acceptable rational approximation of a strongly continuous semigroup $e^{tA} $ on a Banach space, then for t bounded, $\| {r^n (hA)} \| \leqq Cn^{{1/2}} $, with certain improvements under additional hypotheses on r. We also discuss the convergence of $r^n (hA)v$ to $e^{tA} v$ as $h \to 0$ under various assumptions on r and v.