Is it possible to get arthritis in your neck or back?

Yes: Yes. Arthritis in the neck is referred to as "cervical spondylosis" and in the low back as "lumbar spondylosis". It can cause neck or back pain and stiffness, but can become particularly problematic if the arthritis narrows a canal through which a nerve travels. That can cause radiating pain, numbness, and/or weakness into the arm(s) or leg(s).