Multiple intriguing problems hover in adversarial training, including robustness-accuracy trade-off, robust overfitting, and gradient masking, posing great challenges to both reliable evaluation and practical deployment. Here, we show that these problems share one common cause – low quality samples in the dataset. We first identify an intrinsic property of the data called problematic score and then design controlled experiments to investigate its connections with these problems. Specifically, we find that when problematic data is removed, robust overfitting and gradient masking can be largely alleviated; and robustness-accuracy trade-off is more prominent for a dataset containing highly problematic data. These observations not only verify our intuition about data quality but also open new opportunities to advance adversarial training. Remarkably, simply removing problematic data from adversarial training, while making the training set smaller, yields better robustness consistently with different adversary settings, training methods, and neural architectures.