Is Hoeffding's bound tight in any way? The inequality:
$$\Pr(\overline X - \mathrm{E}[\overline X] \geq t) \leq \exp \left( - \frac{2n^2t^2}{\sum_{i=1}^n (b_i - a_i)^2} \right) $$
Is this bound (or any other form of hoeffding) tight in any sense? e.g. does there exist a distribution for which the bound is no more than a constant multiple of the true probability for every $n$?

Yes, one such case is when all the random variables are uniform in $[0,1]$. The sum of uniform random variables has the Irwin-Hall distribution and its tail bound matches the one given by Hoeffding's inequality. See Corollary 5 here.  Also see Bernstein's inequality which is a generalization of Hoeffding.
