Let $X$ be a projective scheme over an artinian commutative ring $R_0$ and let $\Cal{F}$ be a coherent sheaf of $\Cal{O}_X$-modules. We give bounds on the so called cohomological deficiency functions $\Delta^i_{X, \Cal{F}}$ and the cohomological postulation numbers $\nu^i_{X, \Cal{F}}$ of the pair $(X, \Cal{F}).$ As bounding invariants we use the "cohomology diagonal" $ \big( h^j_{X, \Cal{F}}(-j) \big)_{j\le i}$ at and below level $i$ and the $i$-th "cohomological Hilbert polynomial" $p^i_{X, \Cal{F}}$ of the pair $(X, \Cal{F}).$ Our bounds present themselves as a quantitative and extended version of the vanishing theorem of Severi\,--\,Enriques\,--\,Zariski\,--\,Serre.