Magnetic resonance imaging (MRI) of thigh and calf muscles is one of the most effective techniques for estimating fat infiltration into muscular dystrophies. The infiltration of adipose tissue into the diseased muscle region varies in its severity across, and within, patients. In order to efficiently quantify the infiltration of fat, accurate segmentation of muscle and fat is needed. An estimation of the amount of infiltrated fat is typically done visually by experts. Several algorithmic solutions have been proposed for automatic segmentation. While these methods may work well in mild cases, they struggle in moderate and severe cases due to the high variability in the intensity of infiltration, and the tissue's heterogeneous nature. To address these challenges, we propose a deep-learning approach, producing robust results with high Dice Similarity Coefficient (DSC) of 0.964, 0.917 and 0.933 for muscle-region, healthy muscle and inter-muscular adipose tissue (IMAT) segmentation, respectively.