What do you mean by 'teed in the additional light using a junction box'? What specific connections did you make?

If you return all connections to their original configuration, do the original lights all work?

To extend the circuit to an additional light, run the cable from the new fixture to an existing fixture, connect the white wires from the incoming cable, light and new cable together and repeat with the black wires. Connect all grounds together.

I couldn't get inside of the "old work" can shroud (working in attic over sloped ceiling), so I added a junction box 12" away, cut the incoming 14/2 wire, and ran it through a junction box, where I brought in a wire powering the new light 8 feet away. So, the original wire is still connected; there's just one more cable in each wire nut within the junction box. I did exactly what you said electrically, but in a new junction box.

You make an interesting point. I didn't disconnect the new light to test my original configuration, since the new light works and hooks through the same wire nuts powering the old fixture (which now doesn't). Maybe there was a loose connection in the original can and I jostled it (scary thought)? I'll post again after checking; I'm in Florida so I usually reserve attic time for very early morning.

Could you post the circuit diagram of the system, identifying each box and the color of each wire?

So, to where do the switches connect? If the three wires from the third can used to go to the fourth, and now they go to the junction box to which the fourth and fifth cans are connected, hot to hot, neutral to neutral, and ground to ground, with no connections to the switch, then I cannot see any reason why that wouldn't work.

Well, it turns out the second bulb did happen to go bad (interesting timing!), so it's just the one can that's out. That can is connected directly to the junction box, and its wires are firmly in the wire nuts that provide power to the new light. There is a plastic strain relief securing the wire to the original can, so I couldn't have caused much motion there (at most a pound or two of pull when cutting). Everything else is downhill from the main attic where I couldn't even reach it. I moved the insulation and checked visually and by feel to ensure there wasn't a second wire for that can. I checked the wire nut connections (untwisted and retwisted to ensure all three wires were in full contact), I'm wondering if there is any way the fixture itself could fail, and how hard it would be to get it out without falling through the sloped ceiling (diagram of similar device below). If there's no hazard, I can live with what I have, but the potential for loose wires really disturbs me. The access panel is blocked, so I would have to lift it out to look at the internal connections. There is some strange workmanship in this house, but since both switches work and there's only one wire to this can, I can't see how I tapped in at the wrong place. What's your opinion of risk?

I started over, connecting ONLY the existing light (original configuration), and then ONLY the new light. The existing light still doesn't work, so I'm chalking this up to another timing coincidence. I disconnected the original until I understand what's wrong with it. One curious point: inside the can (photo posted below), the neutral wire goes straight to the socket, but the hot wire takes a detour. Could there be an overtemp sensor in the can that went bad? Does this look normal to you? [THANKS for taking the time to help out -- I'm at least confident now that there is no safety hazard.)

CONCLUSION: After ruling out everything else, I've concluded that the original can light failed. I assume this has something to do with the thermal protection sensor. Thanks for your assistance. I've relied on this forum for years and it always delivers. Thank you!!

Thanks for updating and ending the mystery...