What happens when hardware tries to draw more power than power supply can provide?

As far as I understand, computers don't always draw the same amount of power from the power supply all the time. There are times when hard drives are on standby and not using as much as when they're spinning and graphics cards save power when not fully utilized.

What happens when, for example, you have 100 hard drives installed in a desktop tower (or a server rack) with let's say 1000 watt PSU, and they're all on standby, and then suddenly some process accesses all of the hard drives and spins them up, drawing more power than PSU can give?

Is there some signal that hard drives send when they think they're not getting enough power? Or does each individual hardware piece ask the PSU if it can provide it X watts of power, and it may say "no, I don't have that available"? Does the motherboard decide if it can negotiate this power request and safely avoid sudden power loss and instant shutdown? Or is the standard protocol in this case to drop dead without trying to avoid this problem?

From my experience with my desktop and a few hard drives and a low power 350W PSU, it would instantly shut down if 5 hard drives were all trying to spin up at the same time. Nothing bad happened, fortunately, but I'd like to know if instant shutdown is an expected and planned reaction of hardware pieces, or just the motherboard (or PSU) freaking out and disabling everything unexpectedly.

To clarify my question: What I'm interested in is why the common result is system shutdown instead of a safe denial of power to the device which would overload the system? USB power management protects against such a scenario, so why doesn't SATA/Molex power cable management logic not have this (or if it does, why it so commonly fails)?

Update after seeing some answers: I'm really surprised there isn't some sort of power management logic built into PSUs like motherboards have for managing USB power distribution. That's what I got from the answers so far. If you know something that says otherwise, please share as an answer.

From an electronics standpoint, once the current drawn from a supply exceeds the supply capacity then the output voltage will suddenly drop. The electronics that requires a particular voltage to work will simply turn off. This is effectively a power brownout.

In the best case, the power supply detects this overload condition and holds itself off for some period of time or tests to see if the load is still there in a safe fashion, holding the power output off until the load is gone.

In the worst case the power supply goes into constant turn-on and brownout cycles and potentially kills itself or one or more of the devices connected.

There is no provision for devices to "request" more power from a power supply except in intelligent devices like USB where power availability was a concern to begin with. A system power supply has absolutely no intelligent electronics.

The power supply detects an overload condition and shuts down. There is no provision to negotiate power requirements.

Nothing bad happened, fortunately, but I'd like to know if instant shutdown is an expected and planned reaction of hardware pieces, or just the motherboard (or PSU) freaking out and disabling everything unexpectedly.

Theory

It is both. It is the PSU freaking out, which is the expected and planned reaction of hardware pieces. Once in "safety shutdown" mode, you need to either disconnect the PSU for a few seconds or reset a trip switch which is usually on the back¹.

Real world

Very cheap PSUs might not trip, and just leave the equipment malfunctioning due to insufficient current. This is some manufacturers' interpretation of "improving" the ATX design, reducing the need for resetting the PSU in case of a "temporary" overload. Real ATX supplies will be able to operate at maximum peak for a short time, this peak being above continuous operation spec, without tripping and without being damaged. Everyone wins. Poorly designed units of certain manufacturers that shall not be named simply never trip; when used reasonably, they are identical to real ATX PSUs and cost less.

When used unreasonably, they might go up in flames, or enter a "spin-up, brown-out, power-down, power-up, spin-up, brown-out..." cycle that occasionally even self-solves and goes on to a proper boot. What is happening here is that the PSU isn't tripping and the equipment is subjected to unplanned wear and tear. In such a case, I'd advise on replacing the PSU altogether. It makes little sense to run through hoops to accommodate what is a misbehaving power supply to begin with. And while you're at replacing the PSU, get a more powerful one, which solves the initial problem.

Hard disks

However, hard disks are a special case, since they are known to have much greater spin-up requirements. So some hard disks (and motherboards) have provisions to handle this by delaying the spin-up, using jumpers that delay the spin-up by a fixed amount of time, or supporting PUIS (also here) or staggered spin-up via backplane signaling. Jumperless solutions require a suitable motherboard, that is capable of sending the appropriate signal to the hard disk (pin 11 of the SATA interface, implemented by WD and others). The software is either left to the user or sometimes implemented in the BIOS.

This has nothing to do with PSUs, but might explain how, in a certain setup, a 350W PSU might not trip when powering eight hard disks whose combined spin-up power is (I'm using a random number) 400W. That's because the full power drain never materializes, and the hard disks go up two at a time, each drawing a huge current for a few fractions of a second before settling to the normal operational current. Replace the motherboard with one that doesn't handle (or hasn't configured) staggered spin-up, and boom.

Updates

To clarify my question: What I'm interested in is why the common result is system shutdown instead of a safe denial of power to the device which would overload the system? USB power management protects against such a scenario [...] I'm really surprised there isn't some sort of power management logic built into PSUs like motherboards have for managing USB power distribution.

USB is a communication standard between devices that are more "intelligent" than what is required from your average hard disk (granted, the computing power on a hard disk is nothing to be sneered at - some of them can run Linux).

But the problems here are many:

• the PSU cannot be sure about who's draining current. One power line might connect to up to four Molex connectors and the 12V/5V lines aren't designed to carry information. It could be done but you'd need to essentially reengineer both the PSU and all hardware likely to need such a feature.
• denying power to one device might defeat the purpose of booting the whole system. Or lead to potentially disastrous results. Think what would happen if a RAID unit booted one (or two!) disk short due to it having been "current denied".
• if the extreme current requirement stemmed from a hardware fault, the whole system is faulting and therefore the current policy of shutting everything down is, to my eyes, the safest line of conduct. Keep in mind that large, too-important-to-fail systems will be built differently and with huge redundancies, so in those scenarios a limited shutdown is also the best response, and it might not even happen because the faulting unit isn't requiring more current but simply not starting at all (circuit protection and breaking directly on all powered parts. In old high-end IBM AS/400 systems you could short circuit a drive and the system would go on working while one drive bay was going up in flames and smoke - I saw it happen. The unit was logically and electrically disconnected from the backplane, but this didn't prevent it from going on burning, of course; but with enough money even that can be prevented).
• on the gripping hand, it's economically unsound - such a smart PSU would cost much more than a dumber, sturdier, more powerful PSU which would be simpler to build and likely to last longer, and would solve the same problem equally well (actually, having more current at its disposal, and working farther from full capacity, it would solve that particular problem better).

(1) I do remember one Hewlett Packard mini desktop which had it on the inside, next to the cable strips. It also had a "power on" green light on the inside. I imagine these are specially built PSUs for some specific arrangement, that then are employed elsewhere. Disconnecting from the wall outlet should be enough to reset the PSU, but if it isn't, before giving it for dead, try checking the internal side. You never know.