With the rise and ever-increasing potential of deep learning techniques in recent years, publicly available medical data sets became a key factor to enable reproducible development of diagnostic algorithms in the medical domain. Medical data contains sensitive patient-related information and is therefore usually anonymized by removing patient identifiers, e.g., patient names before publication. To the best of our knowledge, we are the first to show that a well-trained deep learning system is able to recover the patient identity from chest X-ray data. We demonstrate this using the publicly available large-scale ChestX-ray14 dataset, a collection of 112,120 frontal-view chest X-ray images from 30,805 unique patients. Our verification system is able to identify whether two frontal chest X-ray images are from the same person with an AUC of 0.9940 and a classification accuracy of 95.55%. We further highlight that the proposed system is able to reveal the same person even ten and more years after the initial scan. When pursuing a retrieval approach, we observe an mAP@R of 0.9748 and a precision@1 of 0.9963. Based on this high identification rate, a potential attacker may leak patient-related information and additionally cross-reference images to obtain more information. Thus, there is a great risk of sensitive content falling into unauthorized hands or being disseminated against the will of the concerned patients. Especially during the COVID-19 pandemic, numerous chest X-ray datasets have been published to advance research. Therefore, such data may be vulnerable to potential attacks by deep learning-based re-identification algorithms.