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.