In this paper, it is proved that the continuity of the tangential components of the average electric and magnetic fields is insufficient to describe the reflection of plane waves by a set of thin parallel wires embedded in a dielectric host using a homogenization approach. It is shown that an additional boundary condition is required to conveniently model a slab of the homogenized metamaterial. In order to understand how the different electromagnetic modes are excited at the interface, the problem of reflection of a plane wave by a set of semi-infinite parallel wires is solved analytically within the thin-wire approximation. Based on the derived result and other arguments, a new boundary condition is proposed for the homogenized wire medium. Extensive numerical simulations support our theoretical analysis, and show that when the additional boundary condition is considered the agreement between full wave results and homogenization theory is very good even for wavelengths comparable with the lattice constant.