As Christian Remling points out, the subtleties appear for $c$ close to $1/2$. That limit can be obtained rather directly from the characteristic function of the <A HREF="https://en.wikipedia.org/wiki/Irwin–Hall_distribution">Irwin-Hall distribution</A>. I find the principal value integral 
$$P_c\equiv\mathbb{P}\left[\sum_{i=1}^n X_i \leq n \cdot c\right]=\frac{1}{2}-\frac{1}{2\pi}\int_{-\infty}^\infty dt\,\frac{1}{(it)^{n+1}}\left(e^{it(1-c)}-e^{-ict}\right)^n$$
or, if we write $c=1/2+\delta$,
$$P_{1/2+\delta}=\frac{1}{2}+\frac{1}{\pi}\int_{0}^\infty dt\,\sin(2nt\delta)\,\frac{\sin^n t}{t^{n+1}}$$

For small $\delta$ this evaluates to

$$P_{1/2+\delta}=\tfrac{1}{2}+2nC_n\delta+{\cal O}(\delta^3)$$

with the coefficient

$$C_n=\frac{1}{\pi}\int_0^\infty dt\,\frac{\sin^n t}{t^n}$$

This integral over the $n$-th power of the sinc function is <A HREF="http://mathworld.wolfram.com/SincFunction.html">well-studied</A>, we need to show that it decreases less rapidly than as $1/n$.