In general, the power rating of speakers is such that when actually using that much power it's pretty stinkin' loud, louder than you'd normally wanna run anyway.
For a given set of speakers, to achieve say, 85 dB at a given listening location, it may require 10 Watts of power from an amplifier, whether that amplifier is 25W, 100W or 2000W.
So if you have a good pair of 40W rated speakers, like my Jamo Cornets, you can safely drive them with a 200W/channel amp, like my Carver M400a, all the live long day because I'll rarely budge the LED display on the amp.
However, if I hook up my Yamaha CR-620 40W/ch to my JBL Northridge E-80s, and tried to watch Pirates of the Carribean at levels which make the cannon fire enjoyable, then the 620 would be pushed beyond its linear range and distort. The distortion components of the amp's signal would overload the high frequency drivers.
On the other hand, if I took the Jamos and the Carver and used them for a PA for a band at the park, I'd have to run more clean power through the speakers than they can take to get it loud enough.
So, it's good to use an amp that will drive your speakers as loud as you'll ever need them to be without distorting. The 2x rule makes sense.
And the distortion is not necessarily from overloading the input as much as the output section running out of gas.
It does what it does. You send it it's full power potential at the input, the output section puts out all that it is designed to.