Steve,
Normally a ArcGISTiledMapServiceLayer is used as the first layer in the map so that it sets the maps LODs (levels of Display) I you are going to add an ArcGISTiledMapServiceLayer at a later point than you need to manually specify the Maps LODs and they will need to be the same (or partially the same) as the ArcGISTiledMapServiceLayer you intend to add. You also need to be concerned with the WKID of all your map layers as the ArcGISTiledMapServiceLayer will not get re-projected to match your WMSLayer.