The Advancement Tests are basically your graduation exam in the game; say you start with two slots in the game. When you pass your first advancement test, you gain a third slot for you to use on your BeastAmulet.
It is only for when you pass your final advancement test, that you cannot do them anymore. You are done with your advancement tests... or are you?
During the battle with Levant, you will begin to realize that that's your TRUE final advancement test, being as you've become a true BeastHunter.
You will also notice that the BeastHunter War option is also gone; that has been removed, and nothing has been put in its place. Instead, the only thing left to do are the Tournaments.