A method is proposed for precisely analyzing the effects of the multiple access interference in the time-hopping ultra-wideband communication systems on the average symbol error rates. An exact analysis is derived for calculating the bit error probability of ultra-wideband systems operating in multiple access interference. The analytical expressions are validated by simulation and used to assess the accuracy of the Gaussian approximation, a widely adopted method for evaluating bit error rate performance. The Gaussian approximation is shown to be inaccurate for predicting bit error rates for medium and large signal-to-noise power ratio values. The different time-hopping modulation schemes are accurately assessed in terms of their sensitivities to the system parameters and the bit error rate.