Yeah, tears (genuine) can be effective. Happened to me in Dallas-FortWorth.
Barbara, one thing you can do, and I recommend you do it, is during your first flight as the stewardess goes by at some point, ask if the flight is on time. Even if it is on time, tell her you've got a close (in time) connecting flight (and have the flight number immediately at hand) and that you're ill/tired and you want to make sure you catch the connection. Ask if the connecting flight can be notified that you are indeed going to be there. They may call ahead for you, so that the second flight will be expecting you at their gate.
Or call the airline a day or so ahead of time and ask them about their policies regarding connecting flights, and let them know your concerns. I've found that when you contact places ahead of time, letting them know you want to be sure you understand something, they try to be helpful. Mainly because you *aren't* a frustrated/angry/hurting customer at that time, I think.