We report experimental results for a boundary-mediated wavenumber-adjustment mechanism and for a boundary-limited wavenumber-band of Taylor-vortex flow (TVF). The system consists of fluid contained between two concentric cylinders with the inner one rotating at an angular frequency Ω. As observed previously, the Eckhaus instability (a bulk instability) is observed and limits the stable wavenumber band when the system is terminated axially by two rigid, non-rotating plates. The band width is then of order ǫ 1/2 at small ǫ (ǫ ≡ Ω/Ωc − 1) and agrees well with calculations based on the equations of motion over a wide ǫ-range. When the cylinder axis is vertical and the upper liquid surface is free (i.e. an air-liquid interface), vortices can be generated or expelled at the free surface because there the phase of the structure is only weakly pinned. The band of wavenumbers over which Taylor-vortex flow exists is then more narrow than the stable band limited by the Eckhaus instability. At small ǫ the boundary-mediated band-width is linear in ǫ. These results are qualitatively consistent with theoretical predictions, but to our knowledge a quantitative calculation for TVF with a free surface does not exist.