I am shocked at the misinformation there is out there about giving a baby a bottle of water. And to my surprise it come from way too many pediatricians. Experts agree that giving water to a baby is not necessary, may even be harmful, before the age of 6 months.
In your average, healthy baby, there is no need whatsoever to give them a bottle of water or sugar water. Ever. 88% of breast milk is water. Even milk made by formula is over 80% water. Even per the American Academy of Pediatrics says "supplements (water, glucose water, formula and other fluids) should not be given to breastfeeding newborn infants unless ordered by a physician when a medical indication exists... During the first 6 months of age, even in hot climates, water and juice are unnecessary for breastfeeding infants and my introduce contaminants and allergens. " The same thing applies to formula fed babies.
Kellymom has some excellent information on this as well here. Introducing water too early can actually cause jaundice or worsen those with existing jaundice, read about that here. When your baby is ill with vomiting and diarrhea and you are scared about them becoming dehydrated, its not the water you need to replace, its the electrolytes. You can re-hydrate using fluids like pedialyte and infalyte. Here is a good link for those preparing milk with formula. Make sure you are preparing the milk appropriately so as to not cause malnutrition.
After your child reaches 6 months old you still need to be cautious when introducing water. You can easily give your child too much and it can be dangerous or down right fatal. For more information click here about water intoxication in infants.