  We study the specification property for partially hyperbolic dynamical
systems. In particular, we show that if a partially hyperbolic diffeomorphism
has two saddles with different indices, and stable manifold of one of them
coincides with the strongly stable leaf, then it does not satisfy the
specification property. As an application, we prove that there exists a
$C^1$-open and dense subset $\mathcal P$ in the set of robustly non-hyperbolic
transitive diffeomorphisms on a three dimensional closed manifold such that
diffeomorphisms in $\mathcal P$ do not satisfy the specification property.
