From what I'm looking up, 13x Jupiter mass to become a brown dwarf star and around 80x Jupiter mass to become a red dwarf star.Sure because the object needed to do that will likely already be a star. Something about 18x the mass of Jupiter, IIRC, is needed to get it to "stardom".
Edit: Which is somewhat surprising to me. As a kid, I had always heard that Jupiter was a "failed star" which meant to me that it was on the brink of becoming a star; just on the threshold and anything large enough crashing into it could cause ignition. In my mind, I thought that if Saturn were ever captured by Jupiter and sucked in, we'd have a second star.