Prior work on exact error rate of diversity receptions over arbitrarily correlated Rician channels involves intractable integrals. Although asymptotic error rates provide closed-form approximations which are accurate at high signal-to-noise ratio, it is still unknown at what signal-to-noise ratio the approximation becomes accurate. In this paper, asymptotically tight closed-form upper and lower bounds are derived for maximal-ratio combining and selection combining over arbitrarily correlated Rician channels. Using these bounds, we can show at what signal-to-noise ratio that the asymptotic error rates are accurate without resorting to time-consuming Monte Carlo simulation.