Pete,
On alt.music.saxophone, as Pete said there was a discussion about this problem and it seems it is quite common and can often be traced back to poor embouchure and/or air column support.
I tried the troublesome G after tuning my sax and it was fine, great! After playing for a little bit I came back to the G and I felt I was on the cusp of it starting again. I realised I was probably 1/4 full on air, if that, and so taking a breath and trying again was much better.
Chris