So can this both mean "They have a big bathroom" and "They have a big bathtub" or just the first one? I can even imagine that this could mean "They have a long bath" or "They have a bath with a lot of water"
This would mean "They have a big bathroom ". 'bathtub' = 'badekar'.
The translation of "bad" to "bath" is confusing to me then, because except in the area of real estate, I've never heard of a bath referring to a bathroom, just to a bathtub.
I still think large is more appropriate to describe the size of a room.