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We derive a Barankin-type bound (BTB) on the mean-square error (MSE) in estimating the directions of arrival (DOAs) of far-field sources using acoustic sensor arrays. We consider narrowband and wideband deterministic source signals, and scalar or vector sensors. Our results provide an approximation to the threshold of the signal-to-noise ratio (SNR) below which the performance of the maximum likelihood estimation (MLE) degrades rapidly. For narrowband DOA estimation using uniform linear vector-sensor arrays, we show that this threshold increases with the distance between the sensors. As a result, for medium SNR values the performance does not necessarily improve with this distance.