What happens when a device draws more current than the power supply can provide?
There are many different things that can happen when you pull too much power and this will be dependent on the technology being used. The typical things you will see is the voltage dropping below the spec'd output or cutting out completely. Some systems might have a fuse that trips when you pull too much power. Or the worst case is you exceed the safe rating of some components that result in something over heating and potentially causes lots of problems (fire, etc).

The circumstances that this can happen pretty much all fall on to the board designer. Many times it is cheaper to build a power supply that can provide less current, so the designer will minimize the cost as much as possible while still being safe. If there is some case that the board designer hasn't considered, then they might pull too much power. These cases might be things like running an RF module while doing some DSP work, while the designer originally intended for them to only operate at separate times.

The other time that you might pull too much power is under an actual fault condition. This is when something truly goes wrong, like an IC fails into a short condition, or someone accidentally shorts two connections together, or something along these lines.

As to what happens to the device, this also all depends on the device itself and the PSU being used. If the PSU drops its voltage, potentially the device might fail if the system can't handle the voltage.