  Let $E/\mathbb{Q}(T)$ be a non-isotrivial elliptic curve of rank $r$. A
theorem due to Silverman implies that the rank $r_t$ of the specialization
$E_t/\mathbb{Q}$ is at least $r$ for all but finitely many $t \in \mathbb{Q}$.
Moreover, it is conjectured that $r_t \leq r+2$, except for a set of density
$0$. In this article, when $E/\mathbb{Q}(T)$ has a torsion point of order $2$,
under an assumption on the discriminant of a Weierstrass equation for
$E/\mathbb{Q}(T)$, we produce an upper bound for $r_t$ that is valid for
infinitely many $t$. We also present two examples of non-isotrivial elliptic
curves $E/\mathbb{Q}(T)$ such that $r_t \leq r+1$ for infinitely many $t$.
