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More likely, you'd be unable to consume the caffeine in a such a diluted form fast enough to exceed the rate at which the liver clears it - it takes time to empty out the stomach of all that water you're consuming along with it. Remember that humans detoxify caffeine much faster than most other mammals, too.

Shouldn't it then tell you - x cups of coffee and you might be dead? It is only LD50, after all.

It's best to be pessimistic in these endeavors.

Pessimistic would be LD1.

At a guess, you'd die from throwing your electrolyte balance off. Just like how drinking too much water can kill you. (Particularly if you drink while sweating - this routinely happens in marathons.)

More likely you'd throw yourself off a bridge, trying to make a record for the longest distance yodelled while falling.