Let $D$ be a simple normal crossing divisor on a smooth projective variety over a field $k \subset \mathcal{C}.$ Write $D_i$ with $i \in I$ for its irreducible components. Denote, as usual,

$D_J=\cap_{i \in J} D_i$

for any subset $J \subset I$. By assumption, all $D_J$ are smooth. Is is true that $H^*(D)$ (Betti or de Rham cohomology) is generated by $H^\ast(D_J)$ for all $J \subset I$?