Alice is being pursued by some Mooks through a high-rise building. She pushes the button for the elevator, but it takes its time getting to her floor. With the mooks catching up to her, she takes the stairs, realizing that by the time the elevator arrives, she could already be captured.
It's common in chase scenes, but a chase isn't necessary for this trope to work. This could come into play during any situation where a character, sick of waiting for an elevator, decides to take the stairs instead.
Note that over one or two floors, stairs are faster even if the elevator is immediately available.
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Anime & Manga
Full Metal Panic? Fumoffu: In a hurry to get up to S˘suke's apartment to retrieve her homework, Kaname first goes to elevator, presses the button, and waits for a few seconds - then screams in frustration and runs for the stairs.
Films — Live-Action
The Ring. Rachel is racing to warn Noah that Samara has not been put to rest and is after him. When she arrives at his apartment she tries to use the elevator, but gets frustrated by the delay and runs up the stairs instead.
In The Blues Brothers, the brothers take the elevator up to the floor the Assessor's office is on, then disable the one elevator they came up in. The combined might of the Chicago Police, SWAT, National Guard, etc. try to follow up the elevator, but after waiting a few seconds they decide to charge up the stairway instead.
In Silk Stockings, in a hotel in Paris, Nina Yashenko, carrying her suitcase, starts walking up several flights of stairs at the same time three of the men she's working with start up in an elevator. She is already waiting there when they arrive.
Star Trek IV: The Voyage Home: Kirk and co. get on the hospital elevator, while the cops chasing them take the stairs. The police are waiting for them when the elevator arrives, only for it to be empty (Kirk and co. beamed to the park).
Human Target: Played Straight. Ames takes the stairs when she realizes Winston is on to her. It doesn't work out for her, though.
24: Jack Bauer proves this trope when he outruns Coffel's elevator, beating him to his limo.
Leverage: In an effort to delay Sterling, who's taking the elevator, Nate runs up the stairs and pushes all the elevator call buttons, causing Sterling's elevator to stop at every floor.
Happens a few times on Alias. Sometimes the stairs aren't necessarily faster, but are stealthier than an elevator.
The Venture Bros.: The Monarch and Henchman 21 take the chute down to the Monarchmobile while Henchman 24 takes the more sensible stairs. He gets there first and closes the sunroof just as they would fall into the car.
An extreme example in Drawn Together: events in the episode had left Captain Hero a quadriplegic. At one point he has to ascend a ludicrously long flight of stairs in his wheelchair. Just a moment after he finally reaches the top, his housemates arrive by elevator.
Parodied in the Futurama episode "Less than Hero", in which Leela descends a set of stairs faster than Fry can fall the same distance. (Since she and Fry, in this episode, can run at super-speed, but not fly, this might even be justified.)
In most cases, this is true so long as it's only three or four floors at the most; the elevator is meant for people carrying packages, for people who may be unable to take stairs due to injury or infirmity, or for people ascending or descending so far that it would make the stairs difficult or impossible. Usually taking the stairs is faster than waiting for the elevator's arrival, boarding the elevator, waiting for the doors to close, waiting for the ascent or descent to the proper floor, potentially being interrupted as new passengers board or exit on the intervening floors, waiting for the doors to open, and disembarking. It's also easier go downstairs than it is to go upstairs.
At the Ohana West Hotel in Waikiki, Hawaii, the stairs are faster, as the lobby only has one or two elevators, one of which is either already in use or out of order.