Well, I can't answer your question either yes or no, since I've yet to see a rheumy for my arthritis. (Don't yet know what type I've got! )
But I don't see why you couldn't have several different things going on at once, such as osteo and rheumatoid arthritis. Especially if one has an autoimmune disease (such as rheumatoid arthritis, for example) it is very common to end up with more than one autoimmune problem.
Hopefully you will receive more replies, from someone who can give you a better answer than mine.
Hugs,