We give strong numerical evidence for the existence of an instability afflicting six-dimensional Reissner-Nordström de Sitter (RNdS) black holes. This instability is akin of the Konoplya-Zhidenko instability present in RNdS black holes in seven spacetime dimensions and above. Moreover, we perform a detailed analysis of the near-horizon limit of extremal RNdS black holes, and find that, typically, unstable gravitational modes effectively behave as a massive scalar field whose mass violates the AdS2 Breitenlöhner-Freedman bound (if and only if d ≥ 6 ), thus providing a physical argument for the existence of the instability. Finally, we show that the frequency spectrum of perturbations of RNdS has a remarkable intricate structure with several bifurcations/mergers that appears unique to RNdS black holes.