  Let $\Gamma$ denote a distance-regular graph with diameter $D\geq 3$.
Juri\v{s}i\'c and Vidali conjectured that if $\Gamma$ is tight with classical
parameters $(D,b,\alpha,\beta)$, $b\geq 2$, then $\Gamma$ is not locally the
block graph of an orthogonal array nor the block graph of a Steiner system. In
the present paper, we prove this conjecture and, furthermore, extend it from
the following aspect. Assume that for every triple of vertices $x, y, z$ of
$\Gamma$, where $x$ and $y$ are adjacent, and $z$ is at distance $2$ from both
$x$ and $y$, the number of common neighbors of $x$, $y$, $z$ is constant. We
then show that if $\Gamma$ is locally the block graph of an orthogonal array
(resp. a Steiner system) with smallest eigenvalue $-m$, $m\geq 3$, then the
intersection number $c_2$ is not equal to $m^2$ (resp. $m(m+1)$). Using this
result, we prove that if a tight distance-regular graph $\Gamma$ is not locally
the block graph of an orthogonal array or a Steiner system, then the valency
(and hence diameter) of $\Gamma$ is bounded by a function in the parameter
$b=b_1/(1+\theta_1)$, where $b_1$ is the intersection number of $\Gamma$ and
$\theta_1$ is the second largest eigenvalue of $\Gamma$.
