Must an uncountable subset of R have uncountably many accumulation points?

Let $T$ be the set of elements of $S$ that are not accumulation points of $S$. Then for each $x \in T$ there exists $\epsilon_x > 0$ such that the interval $I(x,\epsilon_x) = (x-\epsilon_x,x+\epsilon_x)$ contains no other points of $S$. The intervals $\{I(x,\epsilon_x/2) | x\in T\}$ are disjoint, and each contains a rational number; so there can only be a countable number of them.

Hence $T$ is countable, and the set of accumulation points of $S$, which contains $S-T$, is uncountable.