Are you using the BUFFER command in ARCGIS9.1??
If so, make sure that you go into the advanced settings and set the environment variables up to apply the buffers to a projection. ARCGIS by default will do all its GIS commands in what is known as 'geographicals' which equals no projection. As as example buffers on a UTM projection should appear as ovals. But if you let ARCGIS set the default they will appear as circles and the area of coverage will be wrong.
I suspect this is whats happening when you say;
The trouble is that the 90 minute range rings between the lat 67 and lat 78 aerodromes are not the correct distance apart on the map due to the projection. ie. I have told it to project a 609nm ring around both airports. The airports are 1144nm apart, but the two 609nm rings do not overlap - they miss by about 30nm.
A good way to check is as Brian Abraham recommends and that is to calculate some points on the circumference and then cross check them against what the buffer command gives you.
Good luck mate