Elastic scattering of two spinless particles with equal or unequal masses is considered in Feynman-diagram models of the Van Hove type with the usual couplings and with arbitrary spectra of masses, spins, and coupling constants, including infinitely large masses and spins. It is shown that if the amplitude defined by analytic continuation of the sum of all lowest-order diagrams with $s$-channel poles has no unphysical or $u$-channel singularities, then for fixed $t$ this amplitude is not bounded by any power of $s$ as $|s|\ensuremath{\rightarrow}\ensuremath{\infty}$ in any infinitely multiply connected domain which excludes only neighborhoods of the poles. Since a dual Born term with no $u$-channel poles can be represented entirely by the sum of $s$-channel pole diagrams (or by the sum of $t$-channel pole diagrams), this bad asymptotic behavior cannot be canceled in the Born approximation in a dual multiparticle theory. Regge asymptotic behavior is also supposed to be associated with duality, and consequently one sees that Feynman-diagram models of the Van Hove type cannot exhibit duality. The result is independent of the order of summation of diagrams. It depends crucially on the requirement that coupling constants be real.