If your algorithm achieves a 1.5 approximation on all but a finite set $S$ of instances, on which your algorithm achieves a 2-approximation, then you could "improve" your algorithm by "hardwiring" the optimal solutions for the instances in $S$ into your algorithm. In short, for theoretical purposes, an algorithm that succeeds on all but a finite set of instances is just as good as an algorithm that always succeeds. Therefore, a theoretically meaningful tight example is actually an infinite family of tight examples. As Yuval says, any infinite family of examples will do, you don't need an example for every instance size.
That being said, most problems allow you to "scale up" a small example into a larger one.