Council chiefs were unable to say how much the claims were worth, but they are understood to be expecting further claims.
The flood prevention scheme was initiated after about 500 properties were flooded in 2000, causing an estimated £25m of damage.
But Bill Sinclair, 77, of St Bernard’s Row, Stockbridge, the project was not necessary on its current scale.
He said: “I had to leave my house for ten days while the pile driving was going on. The whole building was shaking. Ornaments were falling off shelves. This wonderful house has sat here for 180 years and within a matter of weeks there were cracks in the walls and I can’t open the door properly. The stress has been unbelievable.”