In this paper, we propose a novel person Re-ID model, Consecutive Batch DropBlock Network (CBDB-Net), to help the person Re-ID model to capture the attentive and robust person descriptor. The CBDB-Net contains two novel modules: the Consecutive Batch DropBlock Module (CBDBM) and the Elastic Loss. In the Consecutive Batch DropBlock Module (CBDBM), it firstly conducts uniform partition on the feature maps. And then, the CBDBM independently and continuously drops each patch from top to bottom on the feature maps, which outputs multiple incomplete features to push the model to capture the robust person descriptor. In the Elastic Loss, we design a novel weight control item to help the deep model adaptively balance hard sample pairs and easy sample pairs in the whole training process. Through an extensive set of ablation studies, we verify that the Consecutive Batch DropBlock Module (CBDBM) and the Elastic Loss each contribute to the performance boosts of CBDB-Net. We demonstrate that our CBDB-Net can achieve the competitive performance on the three generic person Re-ID datasets (the Market-1501, the DukeMTMC-Re-ID, and the CUHK03 dataset), three occlusion Person Re-ID datasets (the Occluded DukeMTMC, the Partial-REID, and the Partial iLIDS dataset), and the other image retrieval dataset (In-Shop Clothes Retrieval dataset).