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 is 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 an NP-hardness result for a version of this problem where we have color constraints on vertices. In fact, we show that this version is NP-hard already in bipartite graphs of maximum degree three. In order to achieve the result, we establish a connection between our problem and the problem of construction of special maximum matchings considered in the Master thesis of the author and defended back in 2003.