We consider a model boundary value problem for a quasilinear elliptic system of equations with Venttsel boundary condition on a plane part of the boundary. We establish the partial regularity of the solution in a neighborhood of the plane boundary part by the method of harmonic approximations and describe a condition guaranteeing the local Hölder continuity of the solution along this boundary part. We prove the Hölder continuity in a neighborhood of a boundary point and describe the possible singular set of the solution. Bibliography: 12 titles