I have had the chat box 'break' (chat icon greys out and become unresponsive) many, many times during multiplayer games, but never because I've gone into Settings (because I practically never do during play).
In my experience it tends to become unresponsive if I click on it too close to the transition between holes. If I'm trying to sneak one quick comment in after an opponent's putt has won him the hole but before the scorecard comes up and I click it a fraction too late then the chat icon is broken by the time we tee off on the next.
That said, sometimes it seems to go down spontaneously as well, but 90% of the time it is between holes for me.
Nigel.