Graphene tunnel junctions are a promising experimental platform for single molecule electronics and biosensing. Ultimately their noise properties will play a critical role in developing these applications. Here we report a study of electrical noise in graphene tunnel junctions fabricated through feedback-controlled electroburning. We observe random telegraph signals characterized by a Lorentzian noise spectrum at cryogenic temperatures (77 K) and a 1/f noise spectrum at room temperature. To gain insight into the origin of these noise features, we introduce a theoretical model that couples a quantum mechanical tunnel barrier to one or more classical fluctuators. The fluctuators are identified as charge traps in the underlying dielectric, which through random fluctuations in their occupation introduce time-dependent modulations in the electrostatic environment that shift the potential barrier of the junction. Analysis of the experimental results and the tight-binding model indicate that the random trap occupation is governed by Poisson statistics. In the 35 devices measured at room temperature, we observe a 20–60% time-dependent variance of the current, which can be attributed to a relative potential barrier shift of between 6% and 10%. In 10 devices measured at 77 K, we observe a 10% time-dependent variance of the current, which can be attributed to a relative potential barrier shift of between 3% and 4%. Our measurements reveal a high sensitivity of the graphene tunnel junctions to their local electrostatic environment, with observable features of intertrap Coulomb interactions in the distribution of current switching amplitudes.