Suppose we have two abelian categories $A$ and $B$. Assume that $A$ has enough injectives. Now consider a sequence of functors $F_0,F_1,F_2,....$.
Such that a short exact sequence in $A$ induces a long exact sequence in terms of $F_i$. Can we then claim that $F_i$ are the right derived functors of the functor $F_0$?
If you assume the long exact sequence obtained is functorial in the short exact sequence, then the collection of functors $F_i$ you describe are called a Delta-functor. Delta functors do not always arise as derived functors. A somewhat trivial counterexample: fix a left exact functor $F$, let $F_0=0$ and $F_n=R^{n-1}F$ for $n\geq 1$.