Hardware calibration is essential to restore the uplink/downlink channel reciprocity for multi-user massive multiple-input multiple-output (MIMO) systems operating in a time division duplexing mode. Unfortunately, due to the associated overhead, calibration cannot be performed frequently; furthermore, any calibration procedure leaves behind a residual mismatch between the uplink and downlink channels. In this paper, we study the effects of these calibration errors on the achievable rates in the downlink of a multi-cell, multi-user, and distributed massive MIMO system. Specifically, we develop accurate, yet simple, lower-bounds on the per-user achievable rate, assuming either zero-forcing (ZF) or matched filtering (MF) are used. We also introduce a performance loss coefficient as a measure of sensitivity of the performance to the calibration errors. Using this measure, we identify the conditions under which ZF precoding is more sensitive to calibration errors than MF. Finally, we consider the robust weighted sum-rate maximization problem to mitigate the degrading effects of non-ideal calibration. Our numerical experiments show that the rate lower-bounds developed in this paper accurately quantify the impacts of non-ideal calibration on performance. Also, the proposed robust beamforming scheme improves the average sum-rate by up to 42% compared with the other available schemes.