jmc

We have: $2<\sqrt{t}<\frac{7}{2}$ so squaring the inequality (which we can do because all the terms in it are positive) gives us $4<t<\frac{49}{4}=12.25$. Therefore, $t$ is an integer between 5 and 12 inclusive, which leaves us with $\boxed{8}$ possible integer values of $t$.