Moving quickly

Sometimes I find that my opponent and I are moving very quickly and
I have tried making my move and leaving the board on screen,
waiting for his reply. But I never get the reply. If I then go back
to "Your Games" I often find that he has made the move after all.

My question is: is this a (rather frequent) coincidence or is the system
unable to deliver the reply if I have kept the board on screen?

If you keep the board on screen, you will have to hit the Refresh
button which will appear in stead of the Move button (or hit the F5
key) to update the screen.