I do think it's possible, yes. My son was dairy but not soy for example. It could, of course, go the other way!
I think it would be easier to isolate if you try one at a time in this case. I'd do just a strict soy elimination (given it sounds like the bigger issue) but allow dairy for a few weeks if you can. Then add it back and see if you get a dramatic response. If you still see issues even after the soy completely clears you could then take dairy. But I think it's much easier to do one at a time. It might be easier to avoid only one completely and I think pinpointing the reaction cause will be easier too.