We consider a model problem in a half-ball for linear and quasilinear elliptic systems of equations with nondiagonal principal matrix. It is assumed that the components of the solution are interconnected by Dirichlet and Neumann type boundary conditions through some matrix on the planar boundary of the half-ball. We establish the Hölder continuity of weak solutions to linear systems and the partial regularity of weak solutions to quasilinear systems. To treat such composite boundary conditions, we apply a modification of the method of A-harmonic approximations adapted to problems under consideration.