Hello, Mel:
In 1974, I started my airline pilot career flying a DC-3 between the three Cayman Islands. I remember the Islands very well, from the air in particular, as you might imagine. I left Cayman Airways in 1987 and joined a major airline in the US where I am still employed.
I hope this gets to you OK. You might want to visit and become a member of a Yahoo! Group called Cayman Connections if you are planning on doing any serious Caymanian Genealogy. I can think of no better online resource, as it is frequented by Caymanians who are interested in Caymanian genealogy.
Tibbetts is indeed a very common name in the Cayman Islands, and I do believe in all cases ultimately harkens back to Cayman Brac. Perhaps the most high profile current attraction on Cayman Brac, especially to divers, is an intentionally sunken Russian Navy warship which has been named the MV Keith P. Tibbetts. I personally knew Capt. Keith, as he was commonly known, and knew his children as well, who are all still alive. His son Parker Tibbetts is a businessman and his son Nathaniel is as well, I believe.
Best of luck in your search for information on the Tibbetts family in Cayman Brac!
http://groups.yahoo.com/group/cayman-connections/
David Black