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The interpretation of medical images is a challenging task, often complicated by the presence of artifacts, occlusions, limited contrast and more. Most notable is the case of chest radiography, where there is a high inter-rater variability in the detection and classification of abnormalities. This is largely due to inconclusive evidence in the data or subjective definitions of disease appearance. An additional example is the classification of anatomical views based on 2D Ultrasound images. Often, the anatomical context captured in a frame is not sufficient to recognize the underlying anatomy. Current machine learning solutions for these problems are typically limited to providing probabilistic predictions, relying on the capacity of underlying models to adapt to limited information and the high degree of label noise. In practice, however, this leads to overconfident systems with poor generalization on unseen data. To account for this, we propose a system that learns not only the probabilistic estimate for classification, but also an explicit uncertainty measure which captures the confidence of the system in the predicted output. We argue that this approach is essential to account for the inherent ambiguity characteristic of medical images from different radiologic exams including computed radiography, ultrasonography and magnetic resonance imaging. In our experiments we demonstrate that sample rejection based on the predicted uncertainty can significantly improve the ROC-AUC for various tasks, e.g., by 8% to 0.91 with an expected rejection rate of under 25% for the classification of different abnormalities in chest radiographs. In addition, we show that using uncertainty-driven bootstrapping to filter the training data, one can achieve a significant increase in robustness and accuracy.