I'm looking for an example that shows that the map $c_1: Pic(X) \rightarrow A_{n-1}(X)$ is in general not injective. Eisenbud/Harris gives an exercise using X a plane cubic nodal in 1.35, but I didn't manage to prove it.
Taking two non isom. line bundles, i.e. $\mathcal{O}(p) \ncong \mathcal{O}(q)$ on X and observing that their divisors of zeros $p$ and $q$, respectively are rational equivalent, would be enough. So if maybe all points are rational equivalent on that node, then we are done, but is that true?