The Lesser Antilles are so fragmented because they were (collectively) colonized or captured by no less than eight different countries. Even today, some of them are colonies belonging to France, Britain, the United States, the Netherlands, and Venezuela.
There are only eight sovereign states among them. Of these, Domenica was formerly French and Trinidad-Tobago formerly Spanish. The rest are British. Barbados and St. Vincent and the Grenadines are in a different island group than the others. So the remaining question is why are the four other former British colonies separate?
The "modest" attempts at federation consisted of two islands pairing up around common sea- or air-ports. E.g. St. Kitts and Nevis, St. Vincent and the Grenadines, and Antigua and Barbuda. That reduced 9 to 6.
The unpaired ones are Barbados, st. Lucia and Grenada. St. Lucia and Grenada was formerly French, and doesn't fit in particularly well with other colonies that were "English all the way," or with Domenica. Barbados has more than five times the combined population of St. Kitts and Nevis, more than twice the combined population of St. Vincent and the Grenadines, and the other two smaller nations in each group perhaps feared that they would be "swamped" if Barbados was the "third."
So the "defragmentation" was limited to three pairings (four if you count Trindiad-Tobago). More probably could not have been achieved.