Would this be a faulty connection? Faulty piece of equipment?

Of those two I'd say that you have them wired incorrectly!

Try this link.

Here is a link that might be useful:

You mean 3-way, not 3-pole.

Just a question: WHY would you want to wire it this confusing way? Why not just go from switch to switch with 3-wire???

If you followed that diagram then that is your problem. You CANNOT wire this with 3-wire between the lights. You either need 4-wire or two cables.
Between the lights (besides the ground) you need the two travelers, neutral and switched hot. That is four conductors.
That diagram is TOTALLY wrong. This is what I hate about online diagrams. Folks take them as gospel regardless of source. Any diagrams like this are drawn by someone trying to be nice, or maybe make a buck. We in the trade do not use diagrams, so there are no official stock drawings like that.

Thanks, I got it fixed with that. I guess the original diagram I based it off of was faulty!

So tell us what was wrong. My guess is that the lamps were connected in series.

My original setup had only the 14/3 going between light 1 & 2 (as shown in the diagram I supplied in my first post). I swapped it out for the two strands of 14/2, and now it works perfectly.

As to why did I wire it this way? Well, we already had the first light in place, and then decided that we should add another light in the short hall at the top of the stairs on the same circuit. I was trying to wire it with the leftover wire I had on hand, and none of the lengths would make it from the first switch to the second switch...

Thank you for the help!

Also, thanks for the clarification on pole vs switch.

For many years now, using the "two strands of 14/2" as you describe has been a code violation. NEC 300.3(B) (2011).

Yes! it feels good finally you have discovered the right concept & design