Late gadolinium enhanced (LGE) magnetic resonance (MR) imaging is widely established to assess the viability of myocardial tissue of patients after acute myocardial infarction (MI). We propose the Cascading Refinement CNN (CaRe-CNN), which is a fully 3D, end-to-end trained, 3-stage CNN cascade that exploits the hierarchical structure of such labeled cardiac data. Throughout the three stages of the cascade, the label definition changes and CaRe-CNN learns to gradually refine its intermediate predictions accordingly. Furthermore, to obtain more consistent qualitative predictions, we propose a series of post-processing steps that take anatomical constraints into account. Our CaRe-CNN was submitted to the FIMH 2023 MYOSAIQ challenge, where it ranked second out of 18 participating teams. CaRe-CNN showed great improvements most notably when segmenting the difficult but clinically most relevant myocardial infarct tissue (MIT) as well as microvascular obstructions (MVO). When computing the average scores over all labels, our method obtained the best score in eight out of ten metrics. Thus, accurate cardiac segmentation after acute MI via our CaRe-CNN allows generating patient-specific models of the heart serving as an important step towards personalized medicine.