Suppose that a countably n-rectifiable set M is the support of a multiplicity-one stationary varifold with a point admitting a flat tangent plane T of density Q≥2. We prove that, under a suitable assumption on the decay rate of the blow-ups towards T, there exists a non-constant Brakke flow starting with M. This shows non-uniqueness of Brakke flow under these conditions, and suggests that the stability of a stationary varifold with respect to mean curvature flow may be used to exclude the presence of flat singularities.