A network seems to be shifted when shown against the background map from BING Maps or OpenStreetMap.
The reason for this shift is the difference in map datum, used by the coordinate system of the network and the coordinate system of the background map (usually Sphere Mercator). For the transformation of the map tiles of the background map from Sphere Mercator via WGS84 to the coordinate system of the network no suitable TOWGS parameter are available in the PRJ file of the coordinate system.
In case of (1), applying TOWGS parameter needs a transformation in two steps via WGS84:
First transform to: Network -> Network settings -> (Scale ->) Coordinate system -> From file: 'Sphere_Mercator / GCS_Sphere' Geographic Coordinate Systems -> World -> WGS 1984.prj Confirm all dialogs with OK.
Then transform to the corresponding coordinate system: Network -> Network settings -> (Scale ->) Coordinate system -> From file: 'GCS_WGS_1984' Projected Coordinate Systems -> Utm -> Other GCS -> (e.g.) Irish_National_Grid_TOWGS.prj Confirm all dialogs with OK.