Is it safe, or even possible, to draw more power from a USB port than it was designed to provide
It might be possible in some cases, and it is probably safe to try.

USB only provides 500 mA maximum, and that is only after requesting and being granted that level of current.  However, the spec does not say a port can't supply more, only the minimum it must supply in various circumstances.  The extra logic to limit the port current only to what was negotiated isn't worth it on systems that have plenty of power availble.  Consider that 500 mA at 5 V is only 2.5 W, which is small fraction of a 150-350 W power supply commonly found in desktop systems.  On such systems, they usually just put a polyfuse in series with each USB power line and call it a day.  If you don't draw more than 500 mA, all will be OK.  Polyfuses don't have tight trip points, so there is usually considerable margin.

Laptops, on the other hand, have limited power and therefore do usually manage their USB ports carefully.  There will always be some margin built in, but don't expect a lot more from a laptop USB port than what you're supposed to get.  The margin will be just enough to guarantee the minimum is met under all conditions.

As for it being safe, probably most of the time.  All these devices can't afford to blow up when something bad happens and power and ground are accidentally shorted at a port.  I'd be surprised if you find a computer that is damaged by that.  It's not that much more expensive to build in some basic protection, the probability of crap happening is high enough, and the cost of damage from it high enough that such protection is worthwhile.

Added:

There has been some comments that claim the USB spec requires a host to limit the current it sources to 5 unit loads (500 mA).  That is not correct.  It was also noted that some hosts blow physical fuses if devices draw too much current.  That may be true of those devices, but then those aren't real USB ports.

From the USB 2.0 spec, section 7.2.1, first bullet, page 171, referring to hosts sourcing current:

... must supply at least five unit loads to each port.

Then in section 7.2.1.2.1 "Over-current protection" on page 173:

... The preset value cannot exceed 5.0 A and must be
sufficiently above the maximum allowable port current such that transient currents (e.g., during power up or
dynamic attach or reconfiguration) do not trip the over-current protector.

And later in the same paragraph:

The over-current limiting mechanism must be
resettable without user mechanical intervention. Polymeric PTCs and solid-state switches are examples of
methods, which can be used for over-current limiting.

So clearly:

A host is not only allowed to provide more than 500 mA, it actually must at least for long enough to ride over connection transients.

The only stated upper limit on current a host can provide is 5 A!

Fuses that blow and have to be replaced are not allowed.