Yes it is possible. Like
orangemittens said, you can edit the bones of a Sim rig as an override. Here are some quick steps on how:
Make a custom rig:
1. Clone the Sim object as an override. You can find it in the debug object list.
2. Click the Rig/Slots tab and add the bones that you want by copying similar bones. Make sure that to give them a unique name and set the parent bone appropriately.
In my example I added: (Pelvis) -> b__Tail__0 -> b__Tail__1 -> b__Tail__2
3. Save your package to both your Documents\Sims 4 Studio\Mods\ folder and your regular mods folder.
Attach a new mesh to the custom rig:
1. Restart Studio and open your CAS mesh package.
2. Export the mesh to a .blend file. With your custom rig in the Studio Mods folder, the blend should have your new bones in the armature in Blender.
3. Add the bone names that you added in step 2 to the vertex groups in Blender and weight your mesh to the new vertex groups.
4. Save your blend and import it into Studio.
Animate the custom rig:
1. Open the .blend you made in the previous steps.
2. Select the armature (rig) and un-hide it if it is hidden.
3. Animate as desired.
4. Save and import into a pose pack or animation override.
Results in game: