The supraspinatus attaches to the greater tubercle of the humerus. The greater tubercle gets its name because it is bigger than the lesser tubercle.
Look at this picture and it might help:
There are a few different factors that will determine how much space under the acromion process you have to move. Obviously, with the greater tubercle being bigger, it will further limit that space - creating more of a likelihood for impingement (subacromial) to occur.
The greater tubercle is moved further under that space when we place the shoulder into internal rotation. You may realize this if you try and raise your arm out to the side or overhead with it internal rotation. Obviously, you cannot raise it as high in this position, as you can if the arm were placed in external rotation. This is because external rotation moves the greater tubercle a little more "out of the way" and allows for movement to take place.
So pressing while moving into internal rotation (like the arnold press) may exacerbate this problem (RC pathologies) and give you some pain. This is another reason why I don't think that upper right rows or behind the neck presses are good to do. It comes down to how the shoulder moves and functions.
patrick