Generalizing the Chinese remainder theorem with fibred products

Question

For a commutative ring $R$ and ideals $I,J\subset R$ with $I+J$ we have an isomorphism $$R/(I\cap J)\cong R/I\times R/J.$$ But I'm wondering how much can be said if $I+J\neq R$. We still have an injection $$R/(I\cap J)\ \hookrightarrow\ R/I\times R/J,$$ but this is not surjective in general. In fact we even have an injection into the fibred product $$R/(I\cap J)\ \hookrightarrow\ R/I\times_{R/(I+J)}R/J,$$ which is the usual Cartesian product if $I+J=R$. My question is whether this latter map is surjective in general? For the few simple examples I checked it is indeed surjective. However, my failed attempts at proving this seem to suggest that it isn't surjective, but I haven't been able to find a counterexample.