Diagnosability is an important metric for measuring the reliability of multiprocessor systems. In this paper, we study the diagnosability of a class of networks, called Two-Matching Composition Networks (2-MCNs), each of which is constructed by connecting two graphs via two perfect matchings. By applying our result to multiprocessor systems, we also compute the diagnosability of folded hypercubes and augmented cubes, both of which belong to two-matching composition networks.