Creating a texture which is a subregion of another

I have a source texture which is 4096x4096, comprising a 16x16 array of textures each of which is 256x256. I would like to be able to treat these components as individual textures (each with their own addressing modes, etc.) I do not want to simply index them using UV manipulation because my particular scenario makes that impossible due to the way UV coordinates are generated from vertex positions (don’t ask, I’m not going to change the algorithm just for this in any case.)

My question: Is it possible to create textures which index into a single texture map like this but retain all the other facets of an individual texture, or do I need to create separate texture instances with their own backing data (and then probably discarding the original tiled texture.) If so, where can I see an example of doing this?