Does BigInteger ever overflows?

Question

The API docs says

All of the details in the Spec concerning overflow are ignored, as BigIntegers are made as large as necessary to accommodate the r

Answer 1

If so why do we let some "type" overflow and some don't?

That entirely depends on how it is backed. If it's backed by for example a plain vanilla primitive int, then it will obviously overflow at Integer.MAX_VALUE. Primitives have clear overflow borders, while those for Objects depends on how they're backed/programmed.