Summary: We consider the initial boundary value problem for the 3D Navier-Stokes equations under a slip type boundary condition. Roughly speaking, we are concerned with regularity results, up to the boundary, under suitable assumptions on the directions of velocity and vorticity. Our starting point is a recent, interesting, result obtained by L. C. Berselli and D. Córdoba [C. R., Math., Acad. Sci. Paris 347, No. 11–12, 613–618 (2009; Zbl 1172.35051)] concerning the “near orthogonal case”. We also consider a “near parallel case”.