First the van rolls, then later the van falls off the bridge. Why doesn't this kick wake up Arthur?
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Generally, a kick in some level wakes you up only if you physically wake up in that level. If you are sleeping in a chair and you were given a kick, your brain wakes you up from the dream because it considers that an emergency situation. The same thing happens when in the beginning Arthur doesn't let Cobb breathe when he's dreaming so that he will wake up. Because if that weren't the case, he would die dreaming - which is why your brain exits you from a dream. One thing to note here is that you can exit a dream when your brain wants to exit the dream.
If physical kick was not given, it seems evident from the movie that you can exit the dream either when you want to exit the dream or when you get killed in the dream.
When the van was falling, simple reason for why physical kick didn't occur is that these people had seat belts. They wanted to be sure that they could only exit the dream after the job has been done. This is why they planned for two kicks, in case one fails (and apparently had seat belts too).
Also, they are heavily sedated - which means they have to accept the kick at the level below to wake up in the higher level because they can't kill themselves in dream (they may enter limbo) nor could they physically wake up in the above level (they had seat belts).
When the music played, Arthur says "it's too soon" and doesn't want to quit the dream so early, which is why he doesn't accept the kick (rather his brain doesn't accept the kick and wants to continue the dream). However, he accepts the second kick, after completing his job by giving every one a kick in his level (after they completed theirs).
Reff : Inception Kicks Explained
Here's the deal @Casebash. In complex multi-level dreams, the sedative is really strong. In this dream state it is not enough that you receive just one kick.
Each sleeping person requires two kicks. One from the current level of the dream and one from the level above. These two kicks need to synchronized. This is done using the music.
The van hitting the edge of the bridge is the a kick. If they had synchronized their fall in the dream levels below, they would have woken up. However they miss this kick.
The next kick is the van hitting the water. They synchronize their inner layers kick with this one and they ride the kicks back up to the level of the van.
The link has a description of the pairs of kicks each one uses to wake up each level.