In this paper we consider the problem of full-duplex bidirectional communication between a pair of modems, each with multiple transmit and receive antennas. The principal difficulty in implementing such a system is that, due to the close proximity of each modem's transmit antennas to its receive antennas, each modem's outgoing signal can exceed the dynamic range of its input circuitry, making it difficult - if not impossible - to recover the desired incoming signal. To address these challenges, we consider systems that use pilot-aided channel estimates to perform transmit beamforming, receive beamforming, and interference cancellation. Modeling transmitter/receiver dynamic-range limitations explicitly, we derive tight upper and lower bounds on the achievable sum-rate, and propose a transmission scheme based on maximization of the lower bound, which requires us to (numerically) solve a nonconvex optimization problem. In addition, we derive an analytic approximation to the achievable sum-rate, and show, numerically, that it is quite accurate.