Can a rotator cuff tear cause arm and hand numbness?

Possible/unlikely: If there is a significant amount of swelling around the nerves that pass through to the arm and hand, a rotator cuff tear could cause those symptoms. It is far more likely that a nerve is being pinched- either in the neck or tight muscles around the neck/shoulder. A physiatrist can diagnose and treat these conditions.

Not usually: The shoulder is usually an 'innocent bystander' when it comes to neurologic problems such as numbness and weakness that is found beyond the shoulder in the arm and forearm. The more likely culprit for these complaints is actually nerve compression emanating from the cervical spine.