Say $X$ and $Y$ are two random variables where $X\in [-\alpha,\alpha]$, $Y\in [-\alpha,\alpha]$ and $Z=X+Y$. Is it possible to find two independent random variables with certain pdf (not necessarily identically distributed) that force $Z$ to be uniformly distributed (i.e. $Z\sim \mathcal{U}[-2\alpha,2\alpha]$)?
As the sum of $N$ random variables with zero mean resembles Gaussian distribution with zero mean, I suspect it is not possible to find two such random variables. Do you know any counterexample?