Training wide and deep neural networks (DNNs) require large amounts of storage resources such as memory because the intermediate activation data must be saved in the memory during forward propagation and then restored for backward propagation. However, state-of-the-art accelerators such as GPUs are only equipped with very limited memory capacities due to hardware design constraints, which significantly limits the maximum batch size and hence performance speedup when training large-scale DNNs. Traditional memory saving techniques either suffer from performance overhead or are constrained by limited interconnect bandwidth or specific interconnect technology. In this paper, we propose a novel memory-efficient CNN training framework (called COMET) that leverages error-bounded lossy compression to significantly reduce the memory requirement for training, to allow training larger models or to accelerate training. Different from the state-of-the-art solutions that adopt image-based lossy compressors (such as JPEG) to compress the activation data, our framework purposely adopts error-bounded lossy compression with a strict error-controlling mechanism. Specifically, we perform a theoretical analysis on the compression error propagation from the altered activation data to the gradients, and empirically investigate the impact of altered gradients over the training process. Based on these analyses, we optimize the error-bounded lossy compression and propose an adaptive error-bound control scheme for activation data compression. We evaluate our design against state-of-the-art solutions with five widely-adopted CNNs and ImageNet dataset. Experiments demonstrate that our proposed framework can significantly reduce the training memory consumption by up to 13.5X over the baseline training and 1.8X over another state-of-the-art compression-based framework, respectively, with little or no accuracy loss.