Yes you can! In fact, these type of service-related disabilities have a name: secondary service-connected disabilities. The regulation, 38 C.F.R.§ 3.310(a), provides:
[D]isability which is proximately due to or the result of a service-connected disease or injury shall be service connected. When service connection is thus established for a secondary condition, the secondary condition shall be considered a part of the original condition.
For secondary service-connected disability claims, as with all service-connected disability claims, medical nexus evidence is the key to success. That is, a VA examiner or a private physician must find, within reasonable medical probability, that the service-connected disability is a cause of the secondary condition.
A clear example is a case where you take pain medication for your service-connected back disability, and this medication causes an ulcer or gastrointestinal disability. Assuming a physician’s opinion supports this theory of causation, the ulcer or gastrointestinal disability would be considered a secondary service-connected disability.
Let’s look at a less obvious example of secondary service-connection. You have a service-connected knee disability, which causes instability during walking. One day you walk up a flight of stairs on your back porch. Due to the knee instability, you fall and fractures some vertebras in your spine. Since the service-connected knee instability is a cause of the fall and subsequent back injury, the back injury would be considered service-connected.