  Let $X_{m,n}$ be the Segre-Veronese variety $\mathbb{P}^m \times
\mathbb{P}^n$ embedded by the morphism given by $\mathcal{O}(1,2)$. In this
paper, we provide two functions $\underline{s}(m,n)\le \bar{s}(m,n)$ such that
the $s^{\mathrm{th}}$ secant variety of $X_{m,n}$ has the expected dimension if
$s \leq \underline{s}(m,n)$ or $ \bar{s}(m,n) \leq s$. We also present a
conjecturally complete list of defective secant varieties of such
Segre-Veronese varieties.
