Steven Benton, SEW incident manager, said the water treatment works were “now stable”.
He added: “Low storage levels from this disruption and high demand mean we cannot pump water to some areas, particularly on higher ground.
“To ensure a stable, continuous flow, we must allow tanks to replenish.”
He added that the company was continuing to deliver bottled water to customers from the priority services register, a free support scheme for people who need extra help.
The bottled water stations that have reopened on Sunday are at Tesco Superstore on Pembury Road and Tunbridge Wells Rugby Club.
They will stay open until 20:00 BST, SEW said.
Mike Martin, Tunbridge Wells MP, wrote on X on Sunday morning that the rugby club station had run out of bottled water.