Let me speak from at least my experience... I'm the 3rd, my father was the 2nd (Jr.) and his father was the 1st or Sr., my son is the 4th... We all have exactly the same name which includes the middle name. So it is a progression of a lineage where a name is passed generation-ally without a break. We all have different nick-names to differentiate ourselves. If my son and his wife have a son and decide to follow the tradition then that child will be the 5th. It is possible to treat it differently, but I believe that is the proper method.
Royalty can do it differently and chooses to append the title with a suffix with every re-usage of a name in a royal line... i.e Charles the First and Charles the Second... which incidentally is where my name was patterned after since my ancestors were British Loyalists whom were loyal to one of those Charles's.