Dependent subsets of embedded projective varieties

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety. Set $n:=\dim (X)$. Let $\rho (X{)}^{\mathrm{\prime }\mathrm{\prime }}$ be the maximal integer such that every zero-dimensional scheme $Z\subset X$ smoothable in $X$ is linearly independent. We prove that $X$ is linearly normal if $\rho (X{)}^{\mathrm{\prime }\mathrm{\prime }}\ge \lceil (r+2)\mathrm{/}2\rceil$ and that $\rho (X{)}^{\mathrm{\prime }\mathrm{\prime }}<2\lceil (r+1)\mathrm{/}(n+1)\rceil$, unless either $n=r$ or $X$ is a rational normal curve.