I booked a flight from PTY to MIA and another one 3 hours later from MIA to ORD because it saved me hundreds of dollars. If we get delayed and miss the flight from MIA to ORD wont they put us on the next flight available? I have missed a few flights on different airlines including Spirit and they have always rebooked me on another flight without charging.
If it was booked as a regular stopover (PTY-MIA-ORD), then the airlines responsibility is to get you from PTY to ORD. If there are any delays (or cancellations), they could potentially re-route you (or even put you on another airline) to get you to your final destination.
Once you break the fare into 2 separate reservations, you're taking responsibility for each component (with great savings comes great responsibility).
If the PTY-MIA flight arrives 6 hours late into MIA, the airline still got you where it promised, albeit a few hours later then was supposed to be. The fact that you have a completely separate reservation for a flight that left 3 hours before from MIA-ORD is of no concern to them (you may get a sympathetic agent who can try to help, but officially you're on your own and considered a no-show for the missed flight).