Deep learning models (with neural networks) have been widely used in challenging tasks such as computer-aided disease diagnosis based on medical images. Recent studies have shown deep diagnostic models may not be robust in the inference process and may pose severe security concerns in clinical practice. Among all the factors that make the model not robust, the most serious one is adversarial examples. The so-called "adversarial example" is a well-designed perturbation that is not easily perceived by humans but results in a false output of deep diagnostic models with high confidence. In this paper, we evaluate the robustness of deep diagnostic models by adversarial attack. Specifically, we have performed two types of adversarial attacks to three deep diagnostic models in both single-label and multi-label classification tasks, and found that these models are not reliable when attacked by adversarial example. We have further explored how adversarial examples attack the models, by analyzing their quantitative classification results, intermediate features, discriminability of features and correlation of estimated labels for both original/clean images and those adversarial ones. We have also designed two new defense methods to handle adversarial examples in deep diagnostic models, i.e., Multi-Perturbations Adversarial Training (MPAdvT) and Misclassification-Aware Adversarial Training (MAAdvT). The experimental results have shown that the use of defense methods can significantly improve the robustness of deep diagnostic models against adversarial attacks.