For MSK, not really no.
For rehabilitation, it depends on where you are practicing. For instance, physiatrists in Quebec typically aren't the MRP and only act as consultants while a GP admits the patient and takes of general medical issues. This is similar to what I've seen in Toronto and possibly the rest of Ontario. In other provinces or centers, physiatrists admit the patients under their name and take care of all the general medical issues.
Something to note is that even in Quebec, physiatrists handle a lot of medical issues outside of the MSK and neuro aspects. Disability is rarely limited to these systems and causes many issues in the cardiovascular, urinary, digestive, pulmonary, endocrine, etc. systems. Therefore, you would be expected to handle these issues to prevent complications. For example, a spinal cord injury specialist will handle UTIs, renal complications, osteoporosis, digestive complications, cardiovascular prevention, etc. while an acquired brain injury specialist needs to be comfortable with endocrine disorders and whatnot.