Some conditional independencies rely upon context. For example, for an undirected graph $(a)-(b)-(c)$, variables $b$ and $c$ may be independent while $a=0$, but $b=c$ when $a=1$. Graphical models will err on the side of less independence, so they would not say that $b$ and $c$ are independent and would not describe the context-specific case where they were.