I would tend to agree that it wouldn’t be ground water, because it is along the back of your house and it just seems more likely that it would be an exterior leak. That, along with the fact that your neighbors haven’t had issues, makes it seem like it’s a run-off problem.
BUT, I know that there were streams that used to run in the H Street area, so it could be. Check out this map and zoom in to see if you’re near one of them: http://www.loc.gov/resource/g3850.ct000745/
Where is your downspout draining? How far away from the house have you diverted the downspout? Where does the pipe that the rainwater diverts to run? It may run right under the floor that’s getting flooded and there might be a crack in the pipe that’s buried and you can’t see. Are the gutters all properly attached to the house? Is the wall behind drywall or can you see the actual foundation/wall? If you can’t see the wall, the water might be coming in there and just pooling in the middle because that’s where the lowest spot in the room is. How is your backyard graded? There are a lot of things that could be causing it.