I'm not sure of your definition of breaking the bank, but I can strongly recommend Turneffe Island Resort in Belize. Excellent accomodations, great food and a top-notch diving operation and experience.
TIR bills itself as a "private Caribbean island" as it's the only entity on its own 14-acre caye about 35 miles off the coast. They pick up and drop off guests in Belize City on Wednesdays and Saturdays.
When we were there last summer, they ran two dive boats two to three times a day. Each boat can hold up to 12 divers, but the norm was more like 6 to 8 and sometimes as few as 3. All of the dive sites are within 5-10 minutes of the boat dock, so surface intervals are back at the resort. The exception is on Tuesdays, when they run their big boat to the Blue Hole, which is about an hour twenty minutes away, so they spend the entire day diving in that area before returning to the resort in time for dinner. They also do a Thursday night dive.
It's not cheap, but considering all expenses are paid before you go--except for drinks, nitrox and (if you need it) rental gear--it's a pretty good value for what you get.
My wife and I spent two weeks there and really enjoyed it. We don't usually vacation at the same spot more than once, but in this case we're definitely going to make an exception.http://www.turnefferesort.com/