  Let $b(k,\theta)$ be the maximum order of a connected bipartite $k$-regular
graph whose second largest eigenvalue is at most $\theta$. In this paper, we
obtain a general upper bound for $b(k,\theta)$ for any $0\leq \theta<
2\sqrt{k-1}$. Our bound gives the exact value of $b(k,\theta)$ whenever there
exists a bipartite distance-regular graph of degree $k$, second largest
eigenvalue $\theta$, diameter $d$ and girth $g$ such that $g\geq 2d-2$. For
certain values of $d$, there are infinitely many such graphs of various
valencies $k$. However, for $d=11$ or $d\geq 15$, we prove that there are no
bipartite distance-regular graphs with $g\geq 2d-2$.
