The Monotone Upper Bound Problem asks for the maximal number M(d, n) of vertices on a strictly increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper bound provided by McMullen's [McMullen 70] Upper Bound Theorem is tight, where Mubt (d, n) is the number of vertices of a dual-tocycl ic d-polytope with n facets. It was recently shown that the upper bound M(d, n) ≤ Mubt (d, n) holds with equality for small dimensions (d ≤ 4 [Pfeifle 04]) and for small corank (n ≤ d + 2 [Gärtner et al. 01]). Here we prove that it is not tight in general: in dimension d = 6, a polytope with n = 9 facets can have Mubt (6, 9) = 30 vertices, but not more than M(6,9) ≤ 29 vertices can lie on a strictly increasing edge-path. The proof involves classification results about neighborly polytopes of small corank, Kalai's [Kalai 88] concept of abstract objective functions, the Holt-Klee conditions [Holt and Klee 98], explicit enumeration, Welzl's extended Gale diagrams [Welzl 01], and randomized, generat ion of instances, as well as nonreal izabi Iity proofs via a version of the Farkas lemma.