Joe,
Welcome to the forum.
The ring Chris refers to near the bottom of the case is called the expansion ring, and it is where the brass gets thinnest and widest. It happens at that location because the brass is thicker below that point and doesn't expand much on firing, so the wall at the expansion ring stretches instead to let the head find the breech.
Occasionally you now see foreign made brass where the thick portion of the brass runs a lot further forward in the case than on typical domestic brass, and those get their expansion ring nearer to mid-case. So this may well be a function of whose brass you have. Even if it's a domestic brand, like Remington, it could have been outsourced to a foreign supplier and have this characteristic. Cutting a case in half lengthwise is the only way I know to check.
In any case, it isn't likely to have anything to do with the Lee die.