Can a story have more than one climax?

Yes, a story can have more than one climax, though normally it only has one. The climax is the high point of tension in a story, the turning point.
We could argue that "The Gift of the Magi" has two turning points. Each character makes a significant sacrifice to buy an expensive Christmas gift for the other. Della sells her most prized possession, her long, beautiful hair, to buy a watch chain for her husband, Jim. Jim sells his beloved watch to buy a comb and brush set for Della. The two climaxes comes as each spouse realizes the gift they bought for the other is useless because they have each sold the object that would have made it valuable. In either case, there could have been a bad reaction: hostility, anger, disappointment, reproach. At both moments, we are at a place of high tension, wondering how each spouse will respond to the surprising situation. The fact that both act with love relieves the tensions and allows for the falling action and resolution.

Yes, there can be more than one climax in a story, whether it's within a short story or a novel. A couple of works that famously have more than one climax are A Song of Ice and Fire and The Wheel of Time.
In The Gift of the Magi, there are two climaxes or plot twists in the story. The first one is the revelation that the wife had cut off her hair in order to by a fob for her husband's watch, only to find out the husband bought her a comb. The second climax is the revelation that the husband had sold his watch in order to buy the comb, therefore the presents they bought for each other are useless.
However, their love becomes deeper after seeing that they both sacrificed something for each other. In this regard, the two plot twists or climaxes are intertwined, and both are essential in order to deliver the hard-punching ending to the story.