Don't know about Disney, but certainly in my country, which is in the Caribbean, it is pronounced CaribbEan (empahsis on the "a" and the "e"). It is named after the Caribs, who were the first inhabitants of the region. To me it makes more sense to pronounce the word "carib" and then add on "bean", rather than say Caribbean with the emphasis in the "i".
But hey, even within our region there are differences in pronounciation, so either way is probably acceptable.