The lack of a full range of motion comes with the territory. You will not suffer any ill effects from it, as your muscles are still functioning and contracting through as much range of motion as possible. It is important, though, that you focus more on moves that target your back and isolate it as much as possible during any portion of the range of motion, rather than using any compound movements that may over emphasize a secondary muscle during any part of the range of motion.
For example, with the reverse grip lat pull-down, your lats are doing the work during the first half of the movement, as you finish at the bottom, though, your lats are working very little and your biceps are taking over a majority of the workload. So with this exercise, you would be fine, because the upper half of the motion is the only part you are capable of anyway.
If you were performing standard grip, shoulder width pull-ups, though, where the brache and biceps start the movement, and the lats help to follow through and finish the movement, you would not get sufficient back work, as the areas of the movement where the terres, romboids, and upper lats are most dominant, is the same area you have trouble reaching.
A good safety exercise that emphasizes the various parts of the back through out the entire range of motion with the bare minimum secondary work, would be machine pull-overs.
Just remember, some people like to talk about how the pro's don't use good form, or a full range of motion, well, it's not from a lack of trying, it's from an overwhelming amount of mass. Yet, they still improve upon their already impressive physiques.
You should be fine. Just train right for your body and it's requirements.
I hope this helped.
