The budget of BAT you set each month comes from your wallet, and is essentially your personal donation to the various websites you use, it's not related to ads. This wallet can run out in the same way any wallet can run out, and you'd have to refill it.
The BAT you can earn from seeing ads is separate and unrelated to that. Maybe it goes into the same wallet but that's not important.
You can do none of them, one of them, the other, or both, but they aren't really two related features. They both just happen to use BAT as their money mechanism.