No, they can't. About four years ago the BBC announced that they would be showing a programme featuring flying penguins, and showed a trailer with very convincing footage to advertise it. It looked so good that some people believed it, and failed to notice the date - 1 April 2008. It was in fact a classic April Fools' Day hoax - you can see it here.
Like all birds penguins do have wings, but they act more as flippers. Penguins swim so well that you could almost describe them as flying underwater rather than swimming. As far as is known from the fossil record, even some remote ancestors of modern penguins were flightless, and unlike most birds that dive after fish for their food, there are signs that they began to adapt to the water a very long time ago, with wings functioning as flippers. These ancestors were often huge, sometimes growing nearly 6 feet tall.