I would have thought it unlikely that the flats would not all share the same neutral.Where do the neutrals come into this? All the earths are on a common bussbar.
However, if they didn't, and if the flats were TN-C-S, then bernard would presumably be right. In that situation each of the flat neutrals (which could, if 'different', be at appreciably different potentials) would be, directly or indirectly, connected together via bonding conductors - so, as bernard said, substantial currents could then flow in those conductors. What is this 'common bus bar' connected to?
Kind Regards, John