Sure it's possible and it has been done in the past, but there are several advantages and disadvantages associated with using a texture as the map:
- You'll need some way to determine specific attributes of parts of the map, e.g. whether some position is actually solid, or for other things, such as destructible terrain, ice, etc.
- Making parts of the map dynamic is rather tricky. So you can't do things such as moving platforms or doors as easy (they'll have to be in a different layer).
- Doing bigger maps can get rather memory consuming. Also consider limited hardware capabilities (maximum width/height for textures and such).
The only games I can think of right now using images as maps are several artillery clones, such as Worms, Gunbound, Warmux and others.
If you don't want to use repetitive tiles in a tile map, I'd suggest you use polygons to describe your map and separated tiles/textures to decorate your map. E.g. your map file defines the solid outlines of your map and where to place art assets for the solid areas to be visible in some way (e.g. as a rock).