It actually almost happened! From 1958 to 1962 the West Indies Federation existed, which although still under the flag of the United Kingdom, was a self governing federation of ten former colonies in the Caribbean.

One of the main reasons for its failure was geography, the Islands are relatively sparse, with Jamaica, the largest member, being hundreds of miles from the nearest other island in the federation. Jamaica was in fact one of the main reasons for the failure of the federation, the Jamaicans believing that the smaller islands would become a drain on the economy and making Jamaica poorer at the same time. This was also occuring at a time of large decolonisation within the Bitish Empire with Cyprus, of similar size, being granted full independence in 1960. This was a path that Jamaicans felt confident in taking and they held a referendum on the matter in 1961 on obtaining independence instead of being a member of the federation which they narrowly won by 54% (off the top of my head). Once Jamaica had decided to leave the union, the premier of Trinidad refused to join after disputes with the remaining 8 on ratios of representation and financial issues, and like Jamaica, gained independence in 1962.

The remaining 8 nations attempted to salvage the union, with Barbados playing the lead nation role, however with the two largest members having left, along with the majority of the population, economic assets and financial power, it became an uphill struggle with distances further complicating matters and making the federation financially unviable. Thus the 8 remaining members of the federation retired to their former colony status and gradually received their independence, with the exception of Montserrat which is still a British overseas territory. The Cayman Islands, and the Turks and Caicos islands were separated from Jamaica on independence, and are still British possessions, as is Anguilla, which were separated from St Kitts and Nevis, which gained independence in 1983.

Any further questions then ask away! I will try and add references when I can access my sources.