Graphene nanopores are receiving great attention due to their atomically thin membranes and intrinsic electrical properties that appear greatly beneficial for biosensing and DNA sequencing. Here, we present an extensive study of the low-frequency 1/f noise in the ionic current through graphene nanopores and compare it to noise levels in silicon nitride pore currents. We find that the 1/f noise magnitude is very high for graphene nanopores: typically two orders of magnitude higher than for silicon nitride pores. This is a drawback as it significantly lowers the signal-to-noise ratio in DNA translocation experiments. We evaluate possible explanations for these exceptionally high noise levels in graphene pores. From examining the noise for pores of different diameters and at various salt concentrations, we find that in contrast to silicon nitride pores, the 1/f noise in graphene pores does not follow Hooge's relation. In addition, from studying the dependence on the buffer pH, we show that the increased noise cannot be explained by charge fluctuations of chemical groups on the pore rim. Finally, we compare single and bilayer graphene to few-layer and multi-layer graphene and boron nitride (h-BN), and we find that the noise reduces with layer thickness for both materials, which suggests that mechanical fluctuations may be the underlying cause of the high 1/f noise levels in monolayer graphene nanopore devices.