  Let the Kneser graph $K$ of a distance-regular graph $\Gamma$ be the graph on
the same vertex set as $\Gamma$, where two vertices are adjacent when they have
maximal distance in $\Gamma$. We study the situation where the Bose-Mesner
algebra of $\Gamma$ is not generated by the adjacency matrix of $K$. In
particular, we obtain strong results in the so-called `half antipodal' case.
