  Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d > 0$. Let
$I_\bullet = \{I_n\}_{n \in \mathbb{N}}$ be a graded family of
$\mathfrak{m}$-primary ideals in $R$. We examine how far off from a polynomial
can the length function $\ell_R(R/I_n)$ be asymptotically. More specifically,
we show that there exists a constant $\gamma > 0$ such that for all $n \ge 0$,
$$\ell_R(R/I_{n+1}) - \ell_R(R/I_n) < \gamma n^{d-1}.$$
