You can definitely sing in any key you want. In fact, it is completely possible that you can sing notes just as high as any tenor would, you just might have to work a bit harder for it. I'm a tenor and my range covers about 3 octaves, from E above male high C and down. I had a vocal teacher who was a baritone or bass with about a 7 octave range (Jaime Vendera).
Even if you ignore ever increasing your range dramatically, it makes no sense that you wouldn't be able to sing in Bm. Even with only a two octave range, you can cover every single note in Bm twice. Now it may be the case that you were working on a specific song with you instructor in the key of Gm, and he doesn't think it would be possible for you to sing the same song up two whole steps to Bm at this stage. Or perhaps he had a particular song in mind that happens to be in Bm with notes that go above your current range. The song could sound drastically different with the notes being sung an octave lower, as baritone or bass singers have a very distinct tone when they sing higher notes that tenors usually can't ever pull off.
At this point i'm guessing what your instructor meant. But those are the examples where a particular song in a particular key may not be possible for you to perform at this time. I would ask your teacher for clarity on this, and hopefully he gives a logical answer because on the surface his point makes no sense.