Often we require classification at a very high precision level, such as 99%. We report that when very different sources of evidence such as text, audio, and video features are available, combining the outputs of base classifiers trained on each feature type separately, aka late fusion, can substantially increase the recall of the combination at high precisions, compared to the performance of a single classifier trained on all the feature types i.e., early fusion, or compared to the individual base classifiers. We show how the probability of a joint false-positive mistake can be upper bounded by the product of individual probabilities of conditional false-positive mistakes, by identifying a simple key criterion that needs to hold. This provides an explanation for the high precision phenomenon, and motivates referring to such feature families as (nearly) independent. We assess the relevant factors for achieving high precision empirically, and explore combination techniques informed by the analysis. We compare a number of early and late fusion methods, and observe that classifier combination via late fusion can more than double the recall at high precision.