We describe a non-Arrhenius mechanism for the slowing down of dynamics that is inherent to the high dimensionality of the phase space. We show that such a mechanism is at work both in a family of mean-field spin-glass models without any domain structure and in the case of ferromagnetic domain growth. The marginality of spin-glass dynamics, as well as the existence of a `quasi-equilibrium regime' can be understood within this scenario. We discuss the question of ergodicity in an out-of equilibrium situation.