My thanks to all the responses to my post. There were many helpful suggestions and considerations.

I am grateful to Vilters. I had misunderstood the purpose of MSR. Setting a low value has now produce a vastly more 'accurate' render. Also, I was unaware that Poser's accuracy would be affected if the Image was not in those 4 particular sizes.

I think Millighost's explanation of the sampling and 'linear interpolation' (I need to research a little more , to understand that) well explains how the inaccuracies occur. I don't mind Poser having limitation, but I like to know what they are.

As an example, I attach a 512*512 uncompressed TIF file.

http://www.tima.uk.com/img/simple_lines.tif

It is entirely black, other than some 1 pixel wide lines (on 2 of the edges, and a cross near the centre).  When adding it as a displacement image, no displacement occurs.
I would be grateful if someone might check, as I am a bit new to using Poser. And may easily have done the wrong thing.