I have not given a lot of thought to this question: It may be very easy or very hard or somewhere in between.
Suppose we have a sequence of modules and morphisms which looks like
$ \ldots \to A_1 \to B_1 \to A_2 \to B_2 \to A_3 \to B_3 \to \ldots$
and the two sequences $\ldots \to A_1 \to A_2 \to A_3 \to \ldots$ and $\ldots \to B_1 \to B_2 \to B_3 \to \ldots$ are exact, where we get from $A_1$ to $A_2$ by going through $B_1$ etcetera. Then what can we say about the exactness of the original sequence?
If the original sequence was a complex, then it would be of the form $$ \dotsb\xrightarrow{g_{i-1}}A_{i-1}\xrightarrow{f_{i-1}}B_{i-1}\xrightarrow{g_i}A_i\xrightarrow{f_i}B_i\xrightarrow{g_{i+1}}A_{i+1}\xrightarrow{f_{i+1}}\dotsb $$ where $g_k\circ f_{k-1}=0$ and $f_k\circ g_k=0$.
Now, consider the complex $$ \dotsb\xrightarrow{\phi_{i-2}}A_{i-1}\xrightarrow{\phi_{i-1}}A_i\xrightarrow{\phi_{i}}A_{i+1}\xrightarrow{\phi_{i+1}}\dotsb\tag{1} $$ where $\phi_k=g_{k+1}\circ f_k$. Then $\phi_k=0$ for each $k$. In particular, if $A_i\neq0$, then $(1)$ is not exact at $i$.