Cut the 2 large limes into pieces; combine with the 2 cups water, and the 1 cup of sugar in a blender container. Cover tightly; blend with several on-off motions for 30 seconds or until limes are coarsely chopped. Do not puree. Strain through a sieve into a large pitcher. Dilute mixture with 1 to 2 cups water to taste. Cover and chill.
Wet the rims of 4 to 6 tall glasses; dip rims in sugar. Fill sugar-rimmed glasses with ice. Add 1/2 to 1 ounce rum to each glass. Pour chilled lime mixture over rum. Add extra quartered limes, if desired. Makes 6 servings.