This question was again asked by @firewns. So, let me put it in another way:
An Arahant is not reborn anywhere in the 31 realms of this world when the physical body of the Arahant dies. Then the suffering (that is mostly experienced in the lower realms) ends permanently. The problem with the higher realms (where there is mostly happiness), is that they have finite lifetimes. One ALWAYS ends up in a lower realm, where there can be unimaginable suffering (the animal realm is one of the better ones).
One is born in this world because one has the wrong perception that one can enjoy those material things that are available in this world and become happy.
It is just like a fish thinking that the bait on the hook looks delicious and will bring happiness to it. The fish does not see the hook and that it will be subjected to much pain (hook tearing its mouth), and eventual death.
In the same way, we (all living beings) think that those rupa in this world can bring us happiness. But just like the fish cannot see the suffering hidden in the bait, we cannot see the suffering hidden in sense pleasures. It is only to the purified mind of a Buddha the true nature of this wider world of 31 realms becomes clear. When one fully grasps that real anicca (and dukkha and anatta) nature of this world, one will not crave for those “pleasurable things” anymore.
That understanding leads to the Arahanthood (via three stages of Sotapanna, Sakadagami, Anagami). When one attains Nibbana, one’s mind will not crave for those rupa which have anicca nature. Thus one’s mind will become FREE. In the language of Abhidhamma, nama (or mind) will stop craving for rupa, and thus namarupa formation will stop. A somewhat deep analysis is at: “Nāma & Rūpa to Nāmarūpa“.
Nibbana can be experienced at various levels before one gets to the ULTIMATE RELEASE at the Arahanthood. Nibbana exists, just not in this world; see, “Nibbana“.