Your spine is a quest item in the Fallout: New Vegas add-on Old World Blues.
Background
The Courier's spine is removed by the Sink Auto-Doc, together with the heart and brain on arrival in the Big MT.
Location
During the quest Old World Blues, the spine appears in The Sink and from this point on, can be returned at any time in the Courier's body (as well as removed again) with the same auto-doc.
Notes
- Despite the fact that according to the game-mechanics it is part of the spine, it cannot be taken from the inventory.
- While the spine is not supposed to be seen in-game, using the BaseID to add it to the inventory and then removing the quest item flag allows dropping the spine. Although the dropped spine lacks physics the world model is correct.
- Returning the spine in the Courier's body will replace the Spineless perk with Reinforced Spine.