Joints and splits
Plastic joints can pull apart. Cast-iron pipes can crack or rust through. Water then runs down the wall, often behind the pipe where it is not obvious from the ground.
Blockages higher up
A blockage near the top can make the gutter overflow. A blockage lower down can force water from a weak joint. Both may produce damp inside at a level that makes the roof look guilty.
Check the discharge point
If the pipe empties into a gully, make sure the gully is clear. Water backing up at ground level is a different problem, but repeated saturation can still affect the wall.