The Bellini and Cellini families do collaborate in making a pair.
"Given any pair, there are sixteen possibilities: the gold casket could have been made by Bellini, a son of Bellini, Cellini, or a son of Cellini, and with each of these four possibilities there were four possibilities for the maker for the silver casket."
And as mentioned before, the members of the Cellini family always put false statements on their caskets, and the members of the Bellini family always put true statements.
Here are a couple of example pairs:
Gold: BOTH CASKETS OF THIS SET WERE MADE BY MEMBERS OF THE CELLINI FAMILY
Silver: NEITHER OF THESE CASKETS WAS MADE BY ANY SON OF BELLINI OR ANY SON OF CELLINI
Gold: IF THIS CASKET WAS MADE BY ANY MEMBER OF THE BELLINI FAMILY THEN THE SILVER CASKET WAS MADE BY CELLINI
Silver: THE GOLD CASKET WAS MADE BY A SON OF BELLINI
1. The gold casket was made by Cellini, the silver casket was made by Bellini.
2. The gold casket was made by Bellini, the silver casket was made by Cellini.