Lower back pain is likely one of the most common ailments most people will experience next to the headache. Almost everyone will have it at sometime or another for a variety of reasons. I'm tall but with a short torso so I'm no stranger to lower back problems.
As a cyclist your situated in an unnatural position which can certainly exacerbate any genetic issues you may already have (some are balanced for it more than others). But the same could be said of an office worker that sits in a chair for 8 hours.
The best remedy in either case is to supplement your riding with a good program of resistance training. The stronger the muscles in your lower back, the less pressure on your spine and connective tissue. Of course this presuppose you've already established the optimal riding position.