What about if we're on the back of the drag curve?
Not sure how much you know about this, sorry if you know this already. As you slow an aeroplane down, the wings have to work harder to keep it in the air. The pilot accomplishes this by pointing the nose of the aircraft slightly upwards, increasing the angle of attack on the wings.
A side-effect of this is that, as the wings are working harder, they also create more drag (known as induced drag). This drag requires lots of thrust to overcome it if you want to remain in level flight. So although it's generally true that the slower you fly, the less power you need, there comes a point where the extra drag from the wings actually means you need more power to remain level if you slow down any more.
Now, imagine you are flying along at a very slow speed, such that you are using lots of power to overcome the induced drag from the wings. To recover from this, you would typically lower the nose (to increase the speed), and reduce the power (to prevent the aircraft from climbing). As the speed builds up, the more normal type of drag (known as profile drag) which we are all familiar with increases, and power would gradually need to be added to prevent the aircraft from loosing height.
So, to answer your question, there are cases where you would be able to reduce the power and increase the speed. As for reducing the power all the way to idle, you certainly wouldn't expect the aircraft to remain level at idle power, but it might be that in recovering to a more normal speed, idle power is used for a short period.
All of this is based on light aircraft, but I don't see any reason why a transport jet would behave any differently. You'd need to replace the word "power" with "thrust" everywhere, and if you actually sat down to do the maths there would be some differences, but the net effect would be the same.
Hope that makes sense - it's very difficult to explain without diagrams and graphs!
FFF
---------------
PS - Slingsby, I think you're simple aerodynamics are a little too simple. When I reduce power in my aircraft, it speeds up slightly and starts to descend, which is not what your simple lesson would suggest at all!