One weak answer i.e. a special case would be that if your abelian category $A$ has global dimension 0 ($Ext^i(X,Y) = 0$ for $i>0$), eg- category of vector spaces, then any complex involving elements of $A$ splits into complexes of cycles, boundary, and homologies.

In case your abelian category has global dimension 1, then complexes in the bounded Derived Category(not necessarily in homotopy category!) of $A$ splits into homologies (with sufficient translations), i.e. 

for every $E \epsilon$ $D^b(A)$, $E=\oplus H^i(E)[-i]$.