There are a lot of intersections where I live that don't cycle through the lights in exactly the same way each time. The specific order in which people get green lights (and the length of time they last) varies based on the time of day, but also based on sensors. Since pedestrians can't have those neat sensors they put in the roads to determine if there are cars waiting, the button is all there is.
In situations like that, I think people tend to push it repeatedly because, for one thing, it's not like it's hurting anyone, and for another, if the button isn't pushed at some intersections, there won't ever be a walk signal at all. They may also be thinking that pushing it more often actually will register more "demand" and make the signal appear faster.
With regard to elevator buttons, there are two I use daily at work. The one inside the parking garage stays open for a loooong time if you leave it to its own devices. The close door button definitely cuts off a good 5 seconds on that one. The one inside my building has the doors open for a much shorter time, so I'm not really sure the button does much of anything. It does seem to close the doors a little faster if you hit immediately upon getting in the elevator, but it doesn't always work.