Let $A$ be a ring (commutative and noetherian if it helps). Suppose we are given an inverse system $M_i$ of complexes of $A$-modules (where $i$ is a natural number), and integers $a<b$ such that for each $i$, the complex $M_i$ has non-zero cohomologies only in degrees $a<j<b$.
Consider the complex $\varprojlim M_i$. Does this complex have bounded cohomology?
Note that I am not assuming that this system satisfy a Mittag-Leffler condition.
No, even for $A=\mathbf{Z}$.
Take your favourite example of a complex of short exact sequences of abelian groups whose projective limit is not exact (see for example this math.stackexchange question).
Now just make a complex of long exact sequences by splicing together the short exact sequences a la
$$\to0\to A\to B\to C\to 0\to 0\to A\to B\to C\to 0\to\cdots$$
and you get failure of exactness of the projective limit all over the place, even though all the sequences in the complex are exact.
