Thanks for the correction on my terminology. Steve is indeed correct about the connection v. stopover issue--for instance, a friend of mine was able to arrange a 23-hour "connection" at NRT so that he could spend the night in the city. This did remind me, though, that it is always worth reading the fare rules on international itineraries, because sometimes one (or more) stopovers will be allowed, and they won't have the 24-hour restriction (on the fare my friend was traveling on, the only city where a stopover was not permitted was NRT).

As far as the number of connections permitted--it depends on the airline and the fare rules. For instance, Continental generally says "unlimited online connections" for its domestic trips while Northwest generally limits valid (domestic) routings to no more than four connections in each direction.