Let {W (i) (t), t a a"e(+)}, i = 1, 2, be two Wiener processes, and let W (3) = {W (3)(t), t a a"e (+) (2) } be a two-parameter Brownian sheet, all three processes being mutually independent. We derive upper and lower bounds for the boundary noncrossing probability P (f) = P{W (1)(t (1)) + W (2)(t (2)) + W (3)(t) + f(t) a parts per thousand currency sign u(t), t a a"e (+) (2) }, where f, u : a"e (+) (2) -> a"e are two general measurable functions. We further show that, for large trend functions gamma f > 0, asymptotically, as gamma -> a, P (gamma f) is equivalent to , where is the projection of f onto some closed convex set of the reproducing kernel Hilbert space of the field W(t) = W (1)(t (1)) + W (2)(t (2)) + W (3)(t). It turns out that our approach is also applicable for the additive Brownian pillow.