Being a junior, or a senior, or the third, or the fourth indicates that you are not the first person in your family to have borne a particular name. A child isn’t named Thomas Jefferson Windham, Junior. He is named Thomas Jefferson Windham and becomes Thomas Jefferson Windham, Junior in virtue of there having already been a Thomas Jefferson Windham in the family after whom the younger one is named. And if a third one is born, Thomas Jefferson Windham, Senior becomes Thomas Jefferson Windham I. By the same token, Thomas Jefferson Windham, Junior becomes Thomas Jefferson Windham II, and the youngest is born—and will stay—Thomas Jefferson Windham III. In short, it’s just a way of distinguishing people with the same name.
People do not “move up the ranks,” then, because death does not eliminate the need to refer to previous bearers of the name. Moreover, the point of giving someone the same name as one of their ancestors is typically to start or participate in a family tradition. If Thomas Jefferson Windham III eventually became a junior and then a senior (or perhaps dropped the suffix altogether upon becoming the only one living), both of these uses would be lost. That said, one may refrain from using the suffix in everyday life and leave it to historians (should any be interested). It all depends on one’s interest in the family tradition.
This is just the standard way things go, however. There are many anomalies. A junior might be treated as the senior by his ancestors if he is of some particular importance. Or a family might employ generational suffixes even when names aren’t exactly the same. Wikipedia has some examples here.