Can a union of convex sets be rewritten as a union of disjoint convex sets

Question

Let $C$ be a compact subset of a finite dimensional Euclidean space. Can the union of finitely many convex sets in $C$ (possibly with nonempty intersection) be rewritten as the union of finitely many disjoint convex sets in $C$?

What if instead of "finitely many", we have "countably many"?

Thanks!