About one-third of Atlantic tropical cyclones that make landfall in the U.S. produce something called a "Predecessor Rainfall Event" (PRE). The PRE is a band of rainfall, sometimes quite heavy, that forms out ahead of an approaching tropical cyclone. PRE's are most common when there is a mid-level wave and strong upper jet positioned ahead of the cyclone, resulting in a narrow band of strong lift. PRE's are very difficult to forecast, and can sometimes lead to significant flooding. The heavy rain and flooding can come as something of a surprise, since most attention is concentrated on the incoming tropical storm or hurricane, rather than areas well inland and far away from the main body of the storm.
Below is a radar picture showing a healthy PRE ahead of Hurricane Arthur on the morning of July 4, 2014.