I usually don't mind "island time" as long as the people we are dealing with are friendly and pleasant, and we also try to be nice in our interactions. We are usually happy to sit back, relax, chat, and enjoy the entire experience.
But we have had very poor service in destinations where the staff were just plain rude or even nasty, and that can be galling when they hold out their hand for payment after providing poor service. It also annoys me when someone is mostly rude and inattentive but then suddenly gives us a big smile becomes friendly just before presenting the bill. This applies both in travel situations and here at home, too.
We have experienced a general attitude of rudeness in several Caribbean destinations in places like St. Lucia, a few parts of the Bahamas, and in the Turks and Caicos.
After our first trip to Provo, many years ago, we were very turned off by the unpleasant attitudes of many of locals we encountered. We loved the diving but we didn't return to the T&Cs for 20 years specifically due to the antipathy we felt toward the local populace. When we finally did return we were surprised to find that many of the locals were much more easy-going and friendly and provided good service. It was still expensive but the the change was so markedly different from our memories that I actually did some online research to try and figure out the reasons for the change.
I learned that the T&C tourism organization had been aware of visitors' perceptions of poor service, bad communications, and unsatisfactory interactions with staff, and so they had launched a campaign to try and make a difference. They brought in an outside firm to help address the problem and assist with educating and training business owners and staff - and to make them aware of the importance of the issue to economic growth in a competitive industry. If I recall correctly, the outside group was from Canada - and I've met a lot of nice people in Canada!
We have also been in situations where the locals were nice and friendly even though they lived in a poor and struggling country - Dominica is one such destination that comes to mind.
One of the things we like about the Caymans is that most of the people we meet are polite and friendly. The service isn't always quick, but again, we usually don't mind as long as we are in a pleasant atmosphere among nice people.
@diversteve mentioned that the service could "drive you to distraction on Brac" if you are the impatient type, but we haven't had any problems. In general, the service in the resorts on the Cayman Sister Islands is usually prompt, efficient, and pleasant, but if you venture away from the resorts then things definitely start to slow down. But that's fine, it's a beautiful place and the people are friendly. We are on vacation after all and it's time to relax!
My only significant service issues in Cayman have been in the airports and with Cayman Airways, but that's a whole separate category of frustration!
And those that say that Americans are impatient, demanding, and self-entitled, have you ever spent any time in the rural areas of the Southern US? If you do then you'd better be prepared to sit down and wait - and let people talk - but as long as everybody is nice (both customers and servers) then it can still be a good experience!