First, if you believe that faulty brakes caused you to rear-end another vehicle, you're going to have to make that claim immediately in order for anyone to believe it. That means you're basically going to have to get out of your vehicle at the scene of the accident and tell everyone around that your brakes failed, i.e. "I kept stepping on the brake and nothing happened!"
Make that type of claim three weeks later, and it's not going to be a very credible one.
If you're claiming that your brakes don't work properly, it won't make much sense to drive the car away from the accident scene. If your claim is a credible one, then from a safety standpoint, continuing to drive the vehicle is out of the question. From a practical standpoint, a "faulty brakes" argument will never be believed if the car is not taken (towed) immediately to a garage to be checked out after the accident. Learn more about gathering evidence after a car accident.
If you're claiming that faulty brakes caused your car accident, have the car checked out immediately. If you wait even a couple of days, then it becomes very difficult to prove exactly what the condition of the brakes was at the moment of the collision.
Ideally, the car should be taken by tow truck from the scene of the accident directly to a mechanic. If, for example, the driver wants the car towed home and says that she will get the car checked out soon, that is not a strong sign that the brakes really failed. Or, if the driver says that she has a friend who can check the brakes, that is also going to spell doom for an argument that the brakes failed.
For someone to have a valid claim that the brakes failed, the cause of the failure must have been relatively obvious. It must have been something that the mechanic can immediately spot, like a broken hose or a sheared-off pin. The more complicated the alleged reason for the brake failure, the less likely it is that anyone will believe it.
The key to this question is, if the brakes really failed, why exactly did they fail? The only type of failure that could ever help the driver escape liability is a sudden, complete, unexpected, unforeseeable failure.
Even a sudden, but partial brake failure might be no defense if, for example, the driver wasn't paying attention or was following too closely. In that case, the plaintiff would argue that a reasonable driver wouldn't have followed so closely and that the real cause of the collision was the driver's inattention (negligence), not the brake failure.
Another example would be if the brakes failed completely or even partially due to lack of maintenance or a failure to repair the brakes. This type of failure is no defense whatsoever. Car owners have a legal responsibility to keep their car in a reasonably good state of repair. In this case, the car owner (and driver, if the driver owned the car) will be found negligent for failing to keep the vehicle in reasonably safe condition. If lack of maintenance is the reason for the failure, this also means that the brakes had been slowly failing. Any reasonably aware driver would have or should have noticed that his brakes were not working so well. Learn more about car accidents caused by negligence.