obviously straight lines on (some) projections are very wonky on a sphere (or geoid), as would be shapes built on those lines. That was not really the question though, the question was that if you have shapes defined by points connected with shortest great circle paths then when (and how) would that fail?

Sibling comment already raised the good point that great circles do not work that well on spheroids and other more complex models of Earth, but I got the impression that it was not what apendleton was referring to