To make the simple syrup, combine the blackberries and sugar in a medium sized saucepan. Mash the blackberries just slightly with a fork to help release the juices. Stir in 1 cup of water, and then bring the mixture to a boil. Reduce the heat to low and simmer for 15 minutes, stirring occasionally.
While the simple syrup is simmering, bring 4 cups of water to a boil in a separate saucepan. Add the tea bags, turn off the heat, and allow the tea bags to steep for 3-5 minutes. Remove the tea bags and discard.
After the simple syrup has finished simmering, turn off the heat and allow it to cool slightly, about 3-4 minutes. Set a fine mesh strainer over the top of a heat proof pitcher (I used a large glass pitcher) and strain the simple syrup, pushing down on the blackberries with a spoon to release all of the juice. Discard the leftover pulp.
Pour the tea into the pitcher with the simple syrup and the remaining 4 cups of cold water. Stir and refrigerate the tea until cold. Serve over ice, or add frozen blackberries to help keep it cold, as well.