  Let $A$ be a Noetherian flat $K[t]$-algebra, $h$ an integer and let $N$ be a
graded $K[t]$-module, we introduce and study "$N$-fiber-full up to $h$"
$A$-modules. We prove that an $A$-module $M$ is $N$-fiber-full up to $h$ if and
only if $\mathrm{Ext}^i_A(M, N)$ is flat over $K[t]$ for all $i\le h-1$. And we
show some applications of this result extending the recent result on squarefree
Gr\"obner degenerations by Conca and Varbaro.
