The Bronx and Manhattan were never connected. They have always been separated by the Harlem River, which branches off of the Hudson River, flows down over the top of Manhattan, and into the East River.
However, there is a part of the Bronx -- the Marble Hill neighborhood -- that used to be in Manhattan. The Marble Hill neighborhood used to be on the northermost tip of Manhattan, just north of the Inwood neighborhood. The part of the Harlem River that flowed past Marble Hill was extremely narrow, making it very dangerous for ships to sail from the Hudson River to the East River via the Harlem River.
So in 1895, a deep, wide trench was completed (called the Harlem River Ship Channel), cutting Marble Hill off of Manhattan and making it into an island. In 1914, the narrow, dangerous stretch of river that separated Marble Hill from the Bronx was filled in, making Marble Hill geographically part of the Bronx. However, to this day, the Marble Hill neighborhood is still officially a part of Manhattan, even though it's in the Bronx now.
If you're interested in learning more about the history of the Marble Hill area, I have included a link to the Wikipedia page about the neighborhood.