We consider a boundary crossing probability of a Brownian bridge B-0 and a piecewise linear boundary function u(t) - gammah(t). The main result of this paper is an asymptotic expansion for gamma --> infinity of the boundary crossing probability that B-0(t) is larger than the piecewise linear boundary function u(t) - gammah(t) for some t. Such probabilities occur for instance in the context of change point problems when the Kolmogorov test is used. Examples are discussed showing that the approximation is rather accurate even for small positive gamma values.