We consider a Bose gas with two-body interactions V(r)=gamma^3 v(gamma r) where v(x) is a given repulsive and integrable potential, while $\gamma$ is a positive parameter which controls the range of the interactions and their amplitude at a given distance. Previously, within the Kac scaling, it has been proved in the literature that, in the limit gamma to 0, the gas still undergoes a Bose-Einstein condensation. This can be easily understood by noticing that for gamma=0, the particles feel a uniform potential which only shifts their kinetic energies by the constant a rho, where rho is the particle density and a is the fixed spatial integral of V(r). For non-zero values of gamma, that simple picture is no longer valid and the existence of a condensate is questionable. In fact, using the Hartree-Fock approximation, we find that the condensate is destroyed by
the repulsive interactions when they are sufficiently long-ranged. More precisely, we show that, for gamma sufficiently small but finite, the off-diagonal part of the one-body density matrix always vanishes at large distances. Our analysis sheds light on the coupling between critical correlations and long-range interactions, which might lead to the breakdown of off-diagonal long-range order. The exact status of that breakdown beyond the Hartree-Fock approximation itself remains an open question.