  Let $X\subset \mathbb {P}^r$ be an integral and non-degenerate variety. Set
$n:= \dim (X)$. Let $\rho (X)''$ 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 $\rho (X)''\ge \lceil (r+2)/2\rceil$
and that $\rho (X)'' < 2\lceil (r+1)/(n+1)\rceil$, unless either $n=r$ or $X$
is a rational normal curve.
