Is it unreasonable for the average height of a population to grow by 7" in twenty generations? I should think so. But if you changed your initial conditions somewhat, maybe less unreasonable.
There are roughly 400 genes known to influence height. Imagine we have a small, isolated population that does not interbreed with other populations -- say on an isolated island. This population's average male height is, say 175 cm for men -- roughly the same as the average American. However the population contains all the alleles neede to generate individuals approacing 7' in height. We then take our population and put them under evolutionary pressure; let's say we shoot everyone who reaches the age of 16 and is below average height. It wouldn't many generations for that population's average height to become quite tall, as "tall genes" begin to predominate.
Let's change that initial condition by stipulating that there are no "tall genes" in the initial population. It's still average height, but maybe it lacks both "tall genes" and "short genes". It would be surprising if the genetic height potential for a newborn changed very quickly, because you've got to wait for a lot of "lucky" mutations and twenty generations is not that long.
Let's go back to our successful initial conditions and change something else. This time the population has all the necessary alleles to produce super-tall people, but it interbreeds extensively with a large external population which is not subject to our culling protocol. Under these conditions the population's height increase will be slow, or non-existent depending on the rate at which individuals interbreed with populations not under pressure.
The bottom line: it depends.