"Water Poisoning", also called "Water Intoxication" or "Electrolyte Imbalance" is a potentially fatal condition wherein your body has WAY too much water in it, and WAY too little sodium, potassium, etc.
As far as I've ever read of it, this usually only happens to people who're doing grueling physical labor and are drinking lots of water, and to folk whacked up on Xtasy and too dense to drink not-water.
Here's an example:
When you're hot and exerting yourself, your body loses electrolytes through your sweat (which is why your sweat tastes salty) along with water. In the normal course of the day, drinking water to keep yourself from drying out would be fine, as the salts and other electrolytes in your food would keep you squared away. However, if you're doing just TONS of sweating (e.g., in a bike race, playing football in The Swamp in August, raving like a champion in a club what puts a gas-fired oven to shame), your body can severely deplete its supply of electrolytes, leading to a whole host of problems (hence, the invention of products like Gator-/Power- ade, etc.) if you don't replace them. If you continue to not-replace your electrolytes at a rate fast enough to offset their loss, and continue to take in water, eventually you'll have an electrolyte imbalance significant enough to have severe consequences.
A severe electrolyte imbalance is often accompanied by a SEVERE headache, which can be followed by unconsciousness, coma, and death.
Bear in mind I'm trying to remember all this from a forum thread on a cross-country bike-riding/racing forum, but let me just put your mind at ease:
You would have to drink an absolutely stupendous amount of water (and be losing electrolytes at a pretty good clip) in order to hurt yourself via electrolyte imbalance. If you're still worried about it, drink the water like your doctor says, and just have some gatorade a couple times a day, possibly to the exclusion of a cup of coffee or a beer, depending on the time. And remember to keep yourself in water and 'lectrolytes when you're workin' hard.
Hope I helped,
~GnSx
EDIT: mbs357, I totally know what you mean with the "heavy drinker" bit. If I'm not nursing (read: inhaling) a diet coke or a bottle of water or a glass of this that or the other, i'm on my way to get one. I always feel thirsty, and if I'm not drinking, I have the urge to.. uh.. be doing something with my mouth anyway (thus, my chewed up fingernails, chewing matchsticks, going through gum at a prodigious rate, etc.).