Yes. Many do this regularly, as it is considerably less expensive than the specific replacement cartridges, and generally foam (like sponges or wool) provides good mechanical and bacterial filtration which is usually all you really need. Any chemical filtration achieved by the cartridges (if they contain carbon or other media) is not usually necessary.
The foam or sponge can be rinsed weekly during the partial water change, and only needs replacement if it wears to the point that it no longer provides the surface area. This can be years. Once the foam or sponge begins to "lose shape" so it no longer provides good blockage of the water and the water can get around it, it is not doing anything and can be replaced.
