$R$ a ring $(1\neq 0)$, $\mathbf{Perf}(R)$ is the category of perfect complexes (of right $R$-modules). 


Suppose that $A\_{\bullet}\rightarrow B\_{\bullet}\rightarrow B\_{\bullet}/A\_{\bullet}$ a short exact sequence in $\mathbf{Perf}(R)$ such that


1. $A\_{\bullet}\rightarrow B\_{\bullet}$ is a cofibration in the projective model structure of chain complexes $\mathbf{Ch}\_{R}$.
2. The homology of the chain complexes $A\_{\bullet}$, $B\_{\bullet}$ and $B\_{\bullet}/A\_{\bullet}$ is concentrated at level $n$ (the same $n$ for all $A\_{\bullet}$, $B\_{\bullet}$ and $B\_{\bullet}/A\_{\bullet}$ complexes).


Here is my question: Is it true that $A\_{\bullet}\oplus B\_{\bullet}/A\_{\bullet} $ is isomorphic to $B\_{\bullet}$ in homotopy category $Ho(\mathbf{Ch}\_{R})$.



Split cofibrations up to quasi-isomorphism
No. You can construct counterexamples by taking projective resolutions of modules in a nonsplit short exact sequence. For example, from the short exact sequence $0\to\mathbb{Z}\stackrel{2}{\to}\mathbb{Z}\to\mathbb{Z}/2\mathbb{Z}\to0$ of abelian groups you get
$$\require{AMScd}
\begin{CD}
&&0&&0&&0\\
&&@VVV&@VVV&@VVV\\
0@>>>0@>>>\mathbb{Z}@>\text{id}>>\mathbb{Z}@>>>0\\
&&@VVV&@VV\pmatrix{1\\2}V&@VV\times2V\\
0@>>>\mathbb{Z}@>>>\mathbb{Z}\oplus\mathbb{Z}@>>>\mathbb{Z}@>>>0\\
&&@VVV&@VVV&@VVV\\
&&0&&0&&0\\
\end{CD}$$


