Let $X$ be a smooth projective surface over $\mathbb{C}$. We know that nef line bundles form a closed convex cone whose interior is the ample cone. My doubt is the other direction, is it possible to have an ample class $C$ such that $C=C_1+C_2$ where neither $C_i$ is nef? In fact I know that $C\in |L|$ where $L$ is very ample.

Edit:

I want $C_i$ to be effective. I am interested in the case when $X$ is the Jacobian of a genus 2 curve.