This achievement can be missed.
After you have repaired the gas leak in the abandoned apartment, you can collect the bag. Before you hand it over to Bluto in the Wasps Nest use your inventory to swap over the metal plates with the jewels. You will find the plates in the rubble below the stairs near the boarding house.
Be very careful and make sure you do this, you won't have another chance to. You will know if you have done this successfully as the bag of jewels will be sitting in your inventory.