  We show that, for any integer $n\ge 3$, there is a prime knot $k$ such that
(1) $k$ is not meridionally primitive, and (2) for every $m$-bridge knot $k'$
with $m\leq n$, the tunnel numbers satisfy $t(k\# k')\le t(k)$. This gives
counterexamples to a conjecture of Morimoto and Moriah on tunnel number under
connected sum and meridionally primitive knots.
