Patient medical data is extremely sensitive and private, and thus subject to numerous regulations which require anonymization before disseminating the data. The anonymization of medical documents is a complex task but the recent advances in NLP models have shown encouraging results. Nevertheless, privacy risks associated with NLP models may still remain. In this paper, we present the main privacy concerns in NLP and a case study conducted in collaboration with the Hospices Civils de Lyon (HCL) to exploit NLP models to anonymize medical data.