Person re-identification (Reid) is now an active research topic for AI-based video surveillance applications such as specific person search, but the practical issue that the target person(s) may change clothes (clothes inconsistency problem) has been overlooked for long. For the first time, this paper systematically studies this problem. We first overcome the difficulty of lack of suitable dataset, by collecting a small yet representative real dataset for testing whilst building a large realistic synthetic dataset for training and deeper studies. Facilitated by our new datasets, we are able to conduct various interesting new experiments for studying the influence of clothes inconsistency. We find that changing clothes makes Reid a much harder problem in the sense of bringing difficulties to learning effective representations and also challenges the generalization ability of previous Reid models to identify persons with unseen (new) clothes. Representative existing Reid models are adopted to show informative results on such a challenging setting, and we also provide some preliminary efforts on improving the robustness of existing models on handling the clothes inconsistency issue in the data. We believe that this study can be inspiring and helpful for encouraging more researches in this direction. The dataset is avaliable on the project website: https://wanfb.github.io/dataset.html.