It is worth noting some very interesting cases when the answer is yes. An amazing [result by Miyata][1] states that if $R$ is Noetherian and commutative, $M,N$ are finitely generated  and $E \cong M\oplus N$, any exact sequence
$ 0 \to M \to E \to N \to 0$ must split! 

This holds true slightly more generally, when $R$ is (not necessarity commutative) module-finite over a Noetherian commutative ring. Also, the statement holds for finitely generated pro-finite groups, see Goldstein-Guralnick, J. Group Theory 9 (2006), 317–322. 
  
Added: in fact, this [paper by Janet Striuli](http://www.math.unl.edu/~jstriuli2/research/module2.pdf) may be useful for you. She addressed the question: if two elements $\alpha, \beta \in \text{Ext}^1(M,N)$ give isomorphic extension modules, how close must $\alpha, \beta$ be? Her Theorem 1.2 extend Miyata's result (let $I=0$). 

 [1]: https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-7/issue-1/Note-on-direct-summands-of-modules/10.1215/kjm/1250524308.full