>I found the solution to be quite simple.  I added a 74HC14 hex buffer
>(Schmidt trigger input) between the PIC and the clock line to all the
>74HC165s (only one gate of the buffer being used - ie. all IC clock pins are
>still being driven in parallel).  The HC buffer has a low output impedance
>of approx 50 ohms which will have improved the rise time relative to direct
>driving by the PIC pin (I wonder what the figure is for the PIC pin output
>impedance?).  This has solved my problem!
>

Glad you found it. I was having a lot of trouble picturing the layout, and
obviously guessed wrong.
I'll return the consulting fee :)

Watch ringing on those lines though, you could get into the area I thought
you were in.