Summary: Inflationary cosmological perturbations of quantum-mechanical origin generically interact with all degrees of freedom present in the early Universe. Therefore, they must be viewed as an open quantum system in interaction with an environment. This implies that, under some conditions, decoherence can take place. The presence of the environment can also induce modifications in the power spectrum, thus offering an observational probe of cosmic decoherence. Here, we demonstrate that this also leads to non Gaussianities that we calculate using the Lindblad equation formalism. We show that, while the bispectrum remains zero, the four-point correlation functions become non-vanishing. Using the Cosmic Microwave Background measurements of the trispectrum by the Planck satellite, we derive constraints on the strength of the interaction between the perturbations and the environment and show that, in some regimes, they are more stringent than those arising from the power spectrum.