Has it been proved that there do not exist nice field structures on $\mathbb{R}^n$ for $n\geq 4$?
The quaternions $\mathbb{H}$ fail due to lack of commutativity and the bicomplex numbers $\mathbb{C}\oplus \mathbb{C}$ fail due to non-zero non-invertible elements.
You are probably referring to the Frobenius theorem that the only associative division algebras that are finite-dimensional as vector spaces over the reals are up to isomorphism the reals themselves, the complex numbers, and the quaternions. If commutativity is added only $\mathbb{R}$ and $\mathbb{C}$ remain.
However, it is essential that we require a division algebra or a field to inherit the vector space (or at least the additive group) structure of $\mathbb{R}^n$. Without that we can put a field structure on any $\mathbb{R}^n$ by mapping it bijectively onto $\mathbb{R}$, and it doesn't even make sense to talk about dimension.