Hi - first time poster, long time reader.
I've been following the ATI discussions with interest as I have similar problems with both my P.C and laptop, as they have ATI cards in them (4870 and 3200HD mobility respectively).
The latest drivers for the 4870, catalyst 9.5, do not fix the display problems (the ATI 3200 drivers don't have a recent update). However, the range of the display problems varies, randomly, from load to load. Sometimes, too infrequent for my liking, everything will load and display properly in Oolite on both machines and it looks glorious - Oolite + Neolite skins + Enhanced Planets = win.
Most of the time the star fields don't display properly (comes in two varities, one where the stars look like squares another where they look like cobra mk IIIs - which can be confusing) and the textures on planets only load at a certain distance from the planet. Sometimes the graphics completely bork and almost all the ships and objects in space are completely shadowed, which isn't playable. Fortunately reloading doesn't take long.
I've no idea why the game sometimes displays properly on an ATI card (I know nothing about code or the workings of drivers) and I can live with how the game displays with the weird square stars most of the time. But I do find myself often just reloading the game over and over, waiting for that magical moment where everything runs properly for reasons which escape me.