
If you don't have enough room to claim your reward, it's gone forever. Doesn't drop on the ground, and it doesn't refresh to let you reclaim. It just moves on to the next achievement under that category.
Just a little bug I think might be safe to keep in mind if you don't want to risk this happening when claiming rewards. I found someone else that had the same issue in another thread. It'd be cool if they could accommodate for this by giving you a chance to properly redeem your now lost item.