The islands of the Caribbean have been continuously inhabited since the beginning of the study period. Archeological evidence points to two distinct migrations into the region from Venezuela and Surinam (starting at Trinidad and spreading northwards) around 250 CE and 650 CE. These were presumably the precursors to two of the three major groups that inhabited the islands in 1490: the Ciboney in western Cuba, the Taino in the remaining Greater Antilles (the "big" islands: Cuba, Hispanola, Jamaica, and Puerto Rico). Thus, if you had ancestors in the Greater Antilles in 1000 CE or 500 CE, you probably also had them in adjacent parts of South America in 500 CE or 1 CE.
During the 13th and 14th centuries, these were joined by the Caribs, who left Venezuela to settle the Lesser Antilles. These were the original pirates of the Caribbean, who preyed on their neighbors to the point that women in Carib settlements tended to speak the Taino language rather than that of the Caribs - i.e. they were mostly captives. Thus, if you had ancestors in the Lesser Antilles in 1490 CE , you probably also had them in Venezuela and the Greater Antilles in 1000 CE .
Following the discovery of America, the Spanish colonized the Islands of the Caribbean, and settled heavily in the Greater Antilles so if your ancestors were in the Caribbean in CE 1800, their ancestors might also have been in Spain in CE 1490.
Having nearly exterminated the aboriginal population, the Spanish supplemented their labor force with slaves from West Africa and Angola (see notes on the transatlantic slave trade), so if your ancestors were in the Caribbean in CE 1800, their ancestors were almost certainly also in West Africa in CE 1490.
An interesting variant are the Garifuna or "Black Carib" people, descendants of the Lesser Antilles, Carib-speaking descendants of Caribs and shipwrecked Africans. The Garifuna were deported to the island of Roatan off Honduras in 1796, and eventually settled in Honduras, Guatemala, and Belize, so if your ancestors were Garifuna in CE 1800, their ancestors were certainly in the Lesser Antilles, Venezuela, and West Africa in CE 1490.
Other colonial powers - Other European powers colonized parts of the Caribbean between 1490 and 1800. These include:
Thus, if your ancestors were in any of these places in 1800, their ancestors could have been from anywhere in the colonizing power in 1490.
Finally the US took over the US Virgin Islands in 1917. Ever since 1492, the Caribbean has been a mixing pot for many other nations, with European pirates, and immigrants from all over the colonial empires of the colonizing powers. Thus, although most Caribbeans today look African or Latino, their ancestors are often from as far away as India, China, or the Middle East. Look to your family folklore for illumination.