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By precisely timing them you can measure/check various facts like distance, diameter and so on. In fact, if you time them precisely from different locations on earth you can determine the shape of the occulting body (e.g. an asteroid occulting a star). And on occasion you can get a 'grazing occultation', for example a star goes behind mountains on the moon resulting in it blinking on and off; observe from multiple latitudes and it's possible to recover the profile of the range.