They can if they do too many dives in rapid succession.

You get the bends by having a lot of nitrogen dissolved on your bloodstream due to the pressure. So it's a function of how much nitrogen you breathe, how much pressure you're breathing it at, and the total amount of blood (and some other tissues) in your body that can absorb that nitrogen. Divers get the bends because they are taking multiple breaths of air under pressure, there is multiple lung volumes of nitrogen cycling though the diver. Whales and other diving animals don't typically "hold" their breath when they dive, but if they did, it would only be 1 breath of air for the entire dive. Air in the lungs is bouyancy they don't want and can potentially injure them when it re-expands. Most marine diving animals will saturate their blood and muscles with oxygen at the surface and then dive and exhale.