Yes, chickens can fly. They don't do it often or for very far for a couple of reasons.
Chickens belong to a group of birds that are ground-oriented, and generally only fly to evade predators and to get into a safe roost. Birds that are similar to chickens in this respect are pheasants, turkeys, quail, grouse, prairie chickens, and others.
Birds that fit into this category generally have "white" muscle comprising the major flight muscles (white meat). This type of muscle allows for a very explosive take-off, and rapid acceleration, but over only a fairly short duration. A pheasant is a good example. It explodes out of cover, accelerates rapidly away, but then will glide back into cover again in only a few hundred yards. I can't find my written resource on this, but I believe it is due to the build up of lactic acid in the flight muscles. This type of flight burns energy very rapidly, and exhausts the bird. If re-flushed, the second flush is much weaker and the bird is much less likely to escape.
Birds of this type use camoflage as their primary defense, and flight is only a last resort. They will also fly up into roosts at night to evade ground predators. They primarily travel by walking or running.
Birds that fly greater distances rely on "dark" muscle comprising their flight muscles (dark meat). An example here would be a duck or goose. These birds use flight as their major means of locomotion. They require a more time to gain the air, and accelerate slower. However, the dark muscle allows them much greater endurance. These birds may be airborn for hours at a time.
Another factor relating to the chicken is the "conditioning" of the bird. Physically, these birds are obese and don't get any exercise. We like this because their meat will be tender. But compare a couch potato to an olympic runner, and ask yourself- "Is a couch potato capable of sustained running?" The couch potato can run, sure, but it won't be pretty. Can the couch potato be physically conditioned to run well? Lack of use causes the flight muscles of the chicken to weaken.
Another "conditioning" factor related to the chicken is that it has been conditioned to no longer fear humans as a predator. Organisms that want to eat you generally don't feed you first. Humans do, but that is not normal predator/prey interaction. Why would they fly to escape us, when we have the chicken feed?
Yes, they can fly. They just don't need to, or want to.
MtnMan