IS ATHLETIC ABILITY GENETIC?
This is a very common question and it's not a fun one to answer. The truth is genetics do play a significant role in the vertical jump and athletic ability in general. Look at 12-year old kids. Some are athletic. Some are not. Obviously their varying genetic gifts make a difference. This is a painful realization, because those who are naturally gifted have to admit they did not have to work as much as other people for their abilities, and those who are not gifted have to accept that they can never be the best leaper or sprinter in the world.

There are a number of physical traits that contribute to athletic ability and are influenced by genetics. I'll discuss them in no particular order.
1. Body type. A long, thin body with long limbs is great for jumping and running. Because the tallest people are often not the most athletic, people tend to be unaware of this advantage. When a longer body segment is rotated with the same speed as a shorter segment, the end of that segment covers more distance, meaning that more velocity is generated. In a jump or a sprint, this means that more height is achieved or more ground is covered. As for a thin body, that's just a matter of weight. The less weight on your body per inch of height, the better off you are, at least from a physics standpoint. Obviously, you need muscle mass to be able to generate powerful movements, so you don't want to be as thin as a rail. Unfortunately, your body's build is not something you can do anything about. You can be muscular and lean, but you may still be short and stocky.
2. Strength. Some people are just naturally strong, and they also get stronger easily. This is the reason why tall people are not usually the most athletic. Their strength relative to their body weight is usually much lower, and they have a much harder time making improvements in this area, which more than compensates for the advantage of height. When you combine height with some decent strength levels, you get athletes like Dwight Howard and LeBron. The area of strength is very trainable. A well-trained person can easily get much stronger than an untrained person who is naturally strong. But the gifted person has more potential if he or she does train.
3. Explosiveness. I define explosiveness as the speed at which muscle tension can be generated. This is determined by the nervous system and the fiber type makeup of muscle fibers. Some people are born with great explosiveness. Others are not. Again, this is a very trainable quality, and an non-gifted athlete can still develop very high levels of explosiveness by improving the nervous system and shifting muscle fibers toward the fast-twitch end of the spectrum. However, once again the gifted person has more potential with training.
4. Springiness. We can all recognize the naturally springy people. They seem to effortlessly glide when they move, just playfully bounding around the court or the field and then suddenly bouncing high up into the air as if gravity wasn't even a factor. Springiness is a highly coveted and also an elusive trait. It is largely the product of something called tendon stiffness. Basically, a stiff tendon is like a strong spring. When a force is placed on it, it does not deform very far and it bounces back very hard. Tendon stiffness is trainable, but not as much as other qualities. Elite high jumpers are the people most blessed with tendon stiffness. An average person, even with good training, will never achieve anything close to the level of stiffness possessed by the highly gifted.
5. Recovery Ability. This is similar to the ability to get stronger, but the idea applies to more than just strength training. Some people can naturally bounce back from workouts easily. Those people are able to do more training and make more gains faster. Recovery ability will fluctuate a lot. For example, a pubescent teen can recover from workouts extremely fast. A 40-year old most likely cannot. Also, the first couple weeks of training may cause extreme soreness and possibly a decrease in performance. After a couple more weeks, the body catches up to the stress and begins to make big improvements. Recovery ability can be improved with proper training, rest, and nutrition. Also, over time one can gradually build up the body's tolerance for training, which will allow for more training and more improvement. Of course, there are limits on this. An athlete may improve from age 20 to 25, but is not going to keep improving til 40.

To sum it up, yes athletic ability is significantly influenced by genetics. And the ceiling is higher for the naturally gifted. However, everyone is capable of significantly improving speed and jumping ability and becoming a good athlete. I believe that the typical high school athlete in America could reach a 40-inch vertical with proper training.