The certified flight instructor (CFI) and student pilot were practicing crosswind landings in a Piper PA-28-140. After completing the preflight checks the CFI experienced difficulty closing the top cabin door latch. After multiple attempts he was able to get the latch to lock. The student then began taxing the airplane, but reported the brakes were, "weak." The airplane was equipped with toe-brakes on the pilot side only, which was occupied by the student. The CFI elected to continue with the flight and they performed three uneventful touch-and-go landings. During the fourth takeoff the door latch released, the CFI immediately retarded the throttle and instructed the student to apply the brakes. The student complied but reported that the brakes were not working. The CFI then attempted to slow the aircraft down by applying the center hand brake, but found it to be ineffective. The CFI applied left rudder in an effort to prevent the airplane from rolling to the end of the runway and past the overrun. The airplane departed the runway to the left, struck a drainage ditch, and rolled down an embankment. During the accident sequence the left landing gear was separated, and the left wing sustained substantial damage. The Owners Handbook for the make and model airplane stated, "The cabin door on the Cherokee is double latched, so the chances of it springing open in flight at both the top and bottom are remote. However, should you forget the upper latch, or not fully engage the lower latch, the door may spring partially open. This will usually happen at take-off or soon afterwards. An open door will not affect normal flight characteristics, and a normal landing can be made with the door open." A Federal Aviation Administration (FAA) inspector observed tire skid marks on the runway surface that corresponded to the location of the airplane as it departed the runway.