Anesthesia is VERY competitive now a days. Being a Caribbean student just makes it all that much harder to get into. Caribbean schools are geared moreso to get you into a Primary care field (Family Practice, Internal Med, Pediatrics, etc). It's not impossible to get into more competitive fields, but in order to do so, you need good grades (very hard to do) on top of Good scores on Step 1 and Step 2 (also very hard to do), as well as getting that extra push by doing your rotations at hospitals where you would like to do your residency. It's not impossible, but it is very difficult to do.