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This paper investigates the impact of delayed channel state information at the transmitter (CSIT) on the MIMO broadcast channel with block diagonalization (BD) preceding. First, an upper bound for the achievable throughput is provided, which shows that BD is more robust to imperfect CSIT than zero-forcing precoding as it has fewer inter-user interfering streams. Due to residual inter-user interference, the throughput of BD still saturates at high SNR, which motivates switching between single-user and multi-user precoding. An accurate closed-form approximation is derived for the achievable throughput of the BD system, which provides guidance on the preferred transmission technique for a given scenario.