This paper deals with the study of Fredholm boundary-value problem for the system of fractional differential equations with Caputo derivative. The boundary-value problem is specified by linear vector functional such that the number of it components does not coincide with the dimension of the system of fractional differential equations. We first considered the general solution of the system of fractional differential equations that consist with the general solution of the associated homogeneous system and the arbitrary particular solution of the inhomogeneous system. The particular solution we found is a solution of the system of linear Volterra integral equations of the second kind with weakly singular kernels. Further, by using the theory of pseudo-inverse matrices, we established conditions that should be imposed on the coefficients of the original problem to guarantee that the indicated boundary conditions are satisfied. Moreover, a family of linearly independent solutions of this boundary-value problem is constructed. The specific examples are provided to verify the effectiveness of the proposed approach.