We have investigated the rheology of concentrated oil-in-water emulsions stabilised by silanised silica nanoparticles. The emulsions behave like highly elastic solids in response to small, uniform strains. They become unstable and begin to break down, however, on yielding. We show that the emulsion elasticity is correlated with the salt concentration in the water and hence the particle aggregation in emulsions at a given drop volume fraction. A supporting observation is that destabilisation is favoured by minimising the attractive interactions between the particles. Microscopic observations revealed that coalesced drops have anisotropic shapes and wrinkled surfaces, direct evidence of the interfacial particle layer acting like a mechanical barrier to bulk emulsion destabilisation.