Hořava–Witten spacetimes necessarily include two branes of opposite tension. If these branes are BPS we are led to a puzzle: a negative tension brane should be unstable as it can lose energy by expanding, whereas a BPS brane should be stable as it resides at a minimum of the energy. We provide a detailed analysis of the energy of such braneworld spacetimes in five dimensions. This allows us to show by a non-perturbative positive energy theorem that Hořava–Witten spacetimes are stable, essentially because the dynamics of the branes is entirely accounted for by the behaviour of the bulk fields. We also perform an ADM perturbative Hamiltonian analysis at quadratic order in order to illustrate the stability properties more explicitly.