Fine-grained annotations---e.g. dense image labels, image segmentation and text tagging---are useful in many ML applications but they are labor-intensive to generate. Moreover there are often systematic, structured errors in these fine-grained annotations. For example, a car might be entirely unannotated in the image, or the boundary between a car and street might only be coarsely annotated. Standard ML training on data with such structured errors produces models with biases and poor performance. In this work, we propose a novel framework of Error-Correcting Networks (ECN) to address the challenge of learning in the presence structured error in fine-grained annotations. Given a large noisy dataset with commonly occurring structured errors, and a much smaller dataset with more accurate annotations, ECN is able to substantially improve the prediction of fine-grained annotations compared to standard approaches for training on noisy data. It does so by learning to leverage the structures in the annotations and in the noisy labels. Systematic experiments on image segmentation and text tagging demonstrate the strong performance of ECN in improving training on noisy structured labels.