There is a large debate on the destabilization mechanism of emulsions. We present a simple technique using mechanical compression to destabilize oil-in-water emulsions. Upon compression of the emulsion, the continuous aqueous phase is squeezed out, while the dispersed oil phase progressively deforms from circular to honeycomb-like shapes. The films that separate the oil droplets are observed to thin and break at a critical oil/water ratio, leading to coalescence events. Electrostatic interactions and local droplet rearrangements do not determine film rupture. Instead, the destabilization occurs like an avalanche propagating through the system, starting at areas where the film thickness is smallest.