In a large pot, boil 4 cups of water. Remove from heat. Add the tea bags. Let steep about 10-15 minutes.
Meanwhile, puree blackberries in a blender or food processor. Strain the blackberry puree into a 2 quart pitcher (discarding the pulp and seeds). Strain the water into the pitcher, discarding the tea bags. Stir in sugar. Top with cold water to fill, stirring to incorporate.