A study of the conductance noise in a two-dimensional electron system (2DES) in Si at low temperatures (T ) reveals the onset of large, non-Gaussian noise after cooling from an equilibrium state at a high T with a fixed carrier density ns. This behavior, which signifies the falling out of equilibrium of the 2DES as T → 0, is observed for ns < ng (ng -glass transition density). A protocol where density is changed by a small value ∆ns at low T produces the same results for the noise power spectra. However, a detailed analysis of the non-Gaussian probability density functions (PDFs) of the fluctuations reveals that ∆ns has a qualitatively different and more dramatic effect than ∆T , suggesting that ∆ns induces strong changes in the free energy landscape of the system as a result of Coulomb interactions. The results from a third, waiting-time (tw) protocol, where ns is changed temporarily during tw by a large amount, demonstrate that non-Gaussian PDFs exhibit history dependence and an evolution towards a Gaussian distribution as the system ages and slowly approaches equilibrium. By calculating the power spectra and higher-order statistics for the noise measured over a wide range of the applied voltage bias, it is established that the non-Gaussian noise is observed in the regime of Ohmic or linear response, i.e. that it is not caused by the applied bias.