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In this paper, multiple access channels are considered in which user data rates increase with the signal-to-noise ratio (SNR). To account for channel estimation errors, a finite- SNR diversity-multiplexing tradeoff framework is formulated. The multiplexing gain is defined for each user as the ratio of the user's data rate to the capacity of an additive white Gaussian noise (AWGN) channel. For a set of user multiplexing gains, a finite-SNR diversity gain is defined by the slope of the common outage probability as a function of SNR for quasi-static fading. These definitions have been used in previous work to characterize a finite-SNR diversity-multiplexing tradeoff in multiple access channels and point-to-point multiantenna channels. Inner and outer bounds on the feasible rate region of a multiple access channel with channel estimation errors are used to obtain estimates of the common outage probability. The outage probability is then used to estimate the diversity gain as a function of user multiplexing gains and SNR. It is shown that a finite-SNR formulation is essential to characterizing the impact of channel estimation errors on the diversity-multiplexing tradeoff.