Channel state information (CSI) is important for achieving large rates in MIMO channels. However, in time-varying MIMO channels, there is a tradeoff between the time/energy spent acquiring CSI and the time/energy remaining for data transmission. This tradeoff is accentuated in the MIMO multiple access channel (MAC) as the number of channel vectors to be estimated increases with the number of users. Furthermore, the problem is inherently coupled with multiuser scheduling. We consider a multiple access block fading channel with coherence time $T$, $n$ independent users, each with one transmit antenna and the same average power constraint, and a base station with $M$ receive antennas and no a priori channel state information. We construct a training-based communication scheme and jointly optimize training and user selection. In the high SNR regime, the sum capacity of the non-coherent SIMO MAC is characterized and used to establish the scaling-law optimality of the proposed scheme. At low SNR, the sum rate of the modified scheme decays linearly with vanishing SNR, which is again the optimal rate of decay. These SNR-asymptotic analyses show that while non-trivial scheduling (i.e. selecting a strict subset of the trained users) is sub-optimal at high SNR, it is necessary at low SNR. We thus observe that scheduling gains importance as SNR is reduced. The asymptotic behaviors of the sum rate and throughput per user under increasing $n$ or $M$ are also explored.