Gluing line bundles

Question

Say that I have a scheme $X$ with an open cover $\{U_i\}$ and I know $\operatorname{Pic}(U_i)$ for all $i$. Is this enough information to say something meaningful about $\operatorname{Pic}(X)$? Certainly any line bundle on $X$ will restrict to line bundles on the $U_i$, so we should get an injection $\operatorname{Pic}(X)\rightarrow \oplus \operatorname{Pic}(U_i)$. But I don't think this is enough information to decide if this map is also surjective, we'd need some information about the overlaps, right? What if instead of an open cover $\{U_i\}$, we knew all the Picard groups for a basis $\{U_i\}$, does that change anything?