It's a 19 sector trip, for each leg of which I have a printed coupon. However, to make the itinerary fit onto the ticket, some of the legs have been 'shortened' (e.g. DFW-JFK-DFW-LAX-SFO all on AA would become DFW-SFO) with the full itinerary printed out separately. So the good news is they can print a ticket with more than 16 sectors.

Makes sense to me, just hope it works!