In Gilberg and Trudinger's book, Elliptic Partial Differential Equations of Second Order, Theorem 12.4 states the following :
Let $\Omega \subset \mathbb{R}^2$, and let $u$ be a bounded $C^2(\Omega)$ solution of $$Lu = au_{xx} + 2bu_{xy} + cu_{yy} = f$$ where $L$ is uniformly elliptic, satisfying also $\lambda(\xi ^2 + \eta^2) \leq a\xi^2 + 2b\xi\eta + c\eta^2 \leq \Lambda(\xi^2 + \eta^2) $ and $\frac{\Lambda}{\lambda} \leq \gamma$ for some $\gamma \geq 1$ , where $\lambda, \Lambda$ are the eigenvalues of the coefficient matrix. Then, for some $\alpha= \alpha(\gamma) >0$, we have $$[u]^{*}_{1,\alpha} \leq C\left(|u|_0 + |\frac{f}{\lambda}|_0^{(2)}\right) $$
The remark is made "The validity of the analogue of Theorem 12.4 for n>2 remains in doubt".
I was wondering if anyone knows if this has proved this since the book was published (1998).
The $C^{1,\alpha}$ estimates from Chapter 12, based on the ideas of quasiconformal maps, are indeed restricted to two dimensions. Safonov showed that for $n>2$ not only $C^{1,\alpha}$ fails, but solutions don't even have to be Lipschitz: the Hölder exponent can be arbitrarily small. His result was published in 1987: precise reference is in Counterexamples in PDE on MathOverflow (you may find the rest of the thread interesting too).
Actually, that's just a reprint of 1983 edition.