  Let $X^{(n,m)}_{(1,d)}$ denote the Segre-Veronese embedding of $\mathbb{P}^n
\times \mathbb{P}^m$ via the sections of the sheaf $\mathcal{O}(1,d)$. We study
the dimensions of higher secant varieties of $X^{(n,m)}_{(1,d)}$ and we prove
that there is no defective $s^{th}$ secant variety, except possibly for $n$
values of $s$. Moreover when ${m+d \choose d}$ is multiple of $(m+n+1)$, the
$s^{th}$ secant variety of $X^{(n,m)}_{(1,d)}$ has the expected dimension for
every $s$.
