Yes, if there is no dissolved oxygen in the water the fish can drown.
Also, in the case of labyrinth fish such as the betta, if prevented from reaching the surface, no matter how much oxygen is present in the water, they can drown.
If a shark is prevented from moving in order to get water ot pass over it's gills it can drown as well.
Best of Luck!
EDIT: "When a fish is taken out of water, and exposed only to air, not to oxygen and hydrogen containing water, its gills are unable to control the oxygen intake, the delicate balance cannot be maintained, and the gills inhale a lethal overdose of oxygen. The fish essentially experiences death by "drowning."
So yeah they can drown.
And not all fish can breath atmosphric air, and most can't do it long term. Labyrinth fish are capable or breathing atmospheric air, but others aren't.
Fish like goldfish and most cat fish are capable of sacrificing stomache capacity in order to gulp air and process it in their capillary rich digestive system much like a set of lungs, but it is hard and wearing on the animal.