Place 2 cups of the pineapple cubes in a blender; add 3 cups of the water. Cover and blend until nearly smooth. Press mixture through a fine-mesh sieve into a large pitcher; discard solids. Repeat, using the remaining 2 cups pineapple cubes and the remaining 3 cups water.
Stir honey and lime juice into strained mixture. Pour into a 1/2-gallon insulated jug. Cover and chill overnight.