In this paper, we consider the maximum $k$-edge-colorable subgraph problem. In this problem we are given a graph $G$ and a positive integer $k$, the goal to take $k$ matchings of $G$ such that their union contains maximum number of edges. This problem is NP-hard in cubic graphs, and polynomial time solvable in bipartite graphs as we observe in our paper. We present two NP-hardness results for two versions of this problem where we have weights on edges or color constraints on vertices. In fact, we show that these versions are NP-hard already in bipartite graphs of maximum degree three. In order to achieve these results, we establish a connection between our problems and the problem of construction of special maximum matchings considered in the Master thesis of the author and defended back in 2003.