Let $X$ be a Banach space having a (unconditional, normalized) Schauder basis $(e_n)_n$. Suppose that $Y$ and $Z$ are (closed) block subspaces of $X$ having normalized block bases (with respect to $(e_n)_n$) $(y_n)_n$ and $(z_n)_n$, respectively. Moreover, suppose that the two sets are linearly independent, i.e., their respective linear spans (not their closures) $Y'$ and $Z'$ are such that $Y'\cap Z'=\{0\}$. When can we conclude that $Y\cap Z=\{0\}$ as well?
I'm interested in sufficient criteria, but an exact criterion (even just for the classical sequence spaces, $c_0$, $\ell^p$) would be great.