We consider multi-user MIMO broadcast channels with limited feedback. A recently proposed linear preceding technique, regularized block diagonalization (RBD), is used to mitigate multi-user interference. We assume that each receiver has estimated channel state information (CSI) via downlink training and independently quantizes its channel by using an efficient channel quantization scheme that we propose in this paper. The transmitter acquires the quantized CSI from each receiver through a noiseless and delayed feedback channel. The achievable rates are studied under these assumptions. We derive an upper bound for the rate loss compared to the case that the transmitter has perfect CSI and quantify the impact of channel estimation errors, quantization errors, and outdated quantized CSI on the rate loss. Furthermore, we provide an expression of the number of feedback bits needed per user to maitain that bound. It is found that the delay of the feedback is the predominant cause of performance degradation in the case of rapidly changing channel impulse response.