This has echoes of Cape Town a couple of years ago. This tweet gives an idea of the effect the prolonged dry weather has had on the largest reservoir in Chennai. However, the climate of this area is one of extremes, so I am not sure how bad this actually is - if the rains arrive soon, they will have flooding!
Looks like overdevelopment has been the big issue here, new housing projects with the loss of natural water storage areas and canals with more flooding in the wet years like 2015.
I think in the case of Cape Town a lot of the problems have to do with general poor planning.
I have attached the Wikipedia climate chart for Chennai which shows a distinct 'monsoon season' from roughly October to December. The rainfall from May to September will come from showers and thunderstorms, but for a city with an annual rainfall of nearly 1400mm there must be other reasons why they are short of water and I think spwill has alluded to most of them.