If you measure the water produced by your Brita you won't find 0 TDS water. They are a passive, one step, mixed bed resin mixed with carbon. I'm sure you would render ion-free water if you ran the water through the Brita a few times.
As Melev stated, demineralized water doesn't have a pleasant taste. Completely demineralized water isn't found in natural sources anywhere on earth, so our taste buds don't recognize DI water as desirable.
The water with the warnings I was speaking of is DI or triple distilled water for lab use where truly pure (inorganic and organic-free) water is required, but you bring up a good point. DI water is higher in bacteria and parasites, as only minerals are removed and organic material will pass through the resin. Bacteria also grows on the anaerobic media further feeding the problem. Unless a post UV unit, ozone, or chlorination is used, the water may not be suitable to drink.
Properly bottled water will eliminate the risk of pathogens, but you will find the popular RO/DI water produced by Coke and Pepsi are re-mineralized. I assume remineralization is for flavour rather than for health concerns.
It is electrolytes that hydrate the body by replacing the salts we lose in our sweat. Electrolytes aren't present in DI water. Gatorade is rich in electrolyte salts, but too much of that is also ill-advised.
IF DI water is stored safely with a means of killing bacteria in place, it is perfect for using for cooking and making coffee etc.
You may also want to aerate the water to remove the Co2 that it has stripped from the air, but this all seems like a lot of trouble when RO water can be diverted before it hits your DI.
The World Health Organization weighs in on the subject here.
http://www.who.int/water_sanitation_health/dwq/nutdemineralized.pdf
