Jupiter and the other giant planets are essentially made of the same stuff as the sun, albeit with a few changes to the basic ingredients beyond hydrogen and helium.  So give them a lot more material to start with, and they could ignite nuclear fusion of their hydrogen to form helium, therefore becoming a star.  But brown dwarfs sit in between giant planets and main-sequence stars, too small and light to burn hydrogen, but possibly heavy enough to burn deuterium via nuclear fusion, when they're about 13 times more massive than our Jupiter.