I hate to tell you, but you cannot (should not) combine bump mapping with normal mapping on the same surface. That's a complete waste of processing power. Normal maps are by far superior to bump maps if you can get them, otherwise bump maps can be your fallback option. But of course you cannot apply both.

Or do you mean a "skin"? Don't call a standard image a "normal map" because it is not and you'll just confuse us. And bumpmapping models is really rare, the standard procedure is to use a normal map.