This paper derives high-SNR asymptotic error rates for communications over fading channels by relating them to the asymptotic property of the channel distribution at deep fading, under mild assumptions. The analysis is based on Tauberian theorem and its underlying mathematical theory called regular variation, and applies to a wide range of channel distributions. It is proved that the diversity order being d and the cumulative distribution function (CDF) of the channel power gain having variation exponent d at 0 imply each other, provided that the instantaneous error rate is upper bounded by an exponential function of the instantaneous SNR. This establishes a sufficient condition for the outage event to dominate the error rate performance. Also, asymptotic error rate expressions are derived for practical cases of instantaneous error rate at high average SNR, and turn out to be related to the CDF of the channel power gain in a much simpler manner than existing results. In addition, the high-SNR asymptotic error rate is also characterized under diversity combining schemes, showing the advantage of the approach used herein for analyzing communication systems involving multiple random variables. Numerical results are shown to corroborate our theoretical analysis.