[CW] Speed vs Bandwidth

[Buck]

For some unknown reason, I have been bugged lately about the theory I
learned 25 years ago that the faster a cw signal is sent, the broader
the bandwidth of the signal.  I didn't get my license by learning the
questions and answers.  While I am not a CET or equivilent, I have
studied electronics and had to get my license by learning theory.  (This
isn't a put-down to those who used the q/a method, its my background so
readers get a feel for my understanding of electronics).  

When I studied it, I had no choice but to accept the theory that the
signal width (band-width) is actually greater, the higher the speed of
the cw. 

In practice, I see that the more power I receive from a station, ie the
greater the signal strength, the greater the tuning band-width that I
can receive some noise of interference, as it may be.  However, in cw,
whether its a carrier or someone sending 60+ wpm, at s-5 or so, I can't
tell the difference between the width of either station.  

Has anyone actually tested this theory on a spectrum analyzer?  Is this
a theory we just have to accept or is there any real basis for the
statement.  You folk might have to forgive me, but I can't immagine that
there is actually a cw speed that could consume the entire 80 meter ham
band.  I can imagine a cw speed that is so fast that on 80 meters it
can't be copied reliably but, and maybe I understand it wrongly, I can't
think of anything that will key on and off so fast that my rig will
actually transmit two or three times the signal width in it's specs.  

The world won't come to an end if I never understand this, but if
someone can explain, I would appreciate it.

Thanks

Buck