Yes, it is possible. However, a great deal of engineering may need to be involved. Adding a full height basement adjacent to a more shallow basement will require underpinning the existing foundation and careful attachment of the new foundation to the old. You will be distrubing the "overdig" area of the existing basement. Of course, this will all depend on your soil conditions, etc. Please consult a reputable contractor to explore the situation. As I mentioned, a soil and structural engineer may need to get involved depending on what your local building code officials will require.