Myocardial infarction (MI) is one of the most prevalent cardiovascular diseases and consequently, a major cause for mortality and morbidity worldwide. Accurate assessment of myocardial tissue viability for post-MI patients is critical for diagnosis and treatment planning, e.g. allowing surgical revascularization, or to determine the risk of adverse cardiovascular events in the future. Fine-grained analysis of the myocardium and its surrounding anatomical structures can be performed by combining the information obtained from complementary medical imaging techniques. In this work, we use late gadolinium enhanced (LGE) magnetic resonance (MR), T2-weighted (T2) MR and balanced steady-state free precession (bSSFP) cine MR in order to semantically segment the left and right ventricle, healthy and scarred myocardial tissue, as well as edema. To this end, we propose the Multi-Sequence Cascading Refinement CNN (MS-CaRe-CNN), a 2-stage CNN cascade that receives multi-sequence data and generates predictions of the anatomical structures of interest without considering tissue viability at Stage 1. The prediction of Stage 1 is then further refined in Stage 2, where the model additionally distinguishes myocardial tissue based on viability, i.e. healthy, scarred and edema regions. Our proposed method is set up as a 5-fold ensemble and semantically segments scar tissue achieving 62.31% DSC and 82.65% precision, as well as 63.78% DSC and 87.69% precision for the combined scar and edema region. These promising results for such small and challenging structures confirm that MS-CaRe-CNN is well-suited to generate semantic segmentations to assess the viability of myocardial tissue, enabling downstream tasks like personalized therapy planning.