Let {X-n, n >= 1} be a sequence of centered Gaussian random vectors in R-d, d >= 2. In this paper we obtain asymptotic expansions (n -> infinity) of the tail probability P{X-n > t(n)} with t(n) is an element of R-d a threshold with at least one component tending to infinity. Upper and lower bounds for this tail probability and asymptotics of discrete boundary crossings of Brownian Bridge are further discussed.