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, 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.

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.