I have two lights that are connected oddly and right now only one of them can be on at a time. I would like to be able to have either, both, or neither on by using the 3 switch positions I already have. How do I connect the switches to make this work? Does one of them need to be replaced with a 4 way switch?
OK first, nobody has a chance to figure out that Egyptian Spaghetti (black red white bronze = Egyptian flag). So let's hit up the electrical supply house for some tape, and re-mark the colors so we can see what's going on.
Normally I use the same color for both messengers, but in your case, it matters.
Normally, a 3-way works by a 3-way connecting always-hot to one of two messengers (the other is dead). Then the other 3-way connects one of two messengers to switched-hot. If the switches agree, switched-hot is energized, and the lamp gets switched-hot and neutral, and lights when switched.
Now, you just horked on another lamp and 3-way, as if this was Insteon or something. Look what you did.
The original 3-way circuit still works normally. Bulb B gets always-hot and a messenger. No neutral in sight - that's a big problem. If switch 1 agrees with 2, you have 120V on both sides of bulb B, and no light.
If switch 1 is the minority report, then it's talking to a dead wire, and no light.
But what happens if switch 2 is the minority report? Follow the loop:
Always Hot - Bulb B - switch 1 - dead messenger - switch 3 - bulb A - neutral
You have put bulbs B and A in series. Now if bulb B is an efficient LED/CFL, and bulb A is an incandescent, A will barely light.
How do you fix this? Easy. Get switched-hot and neutral to bulb B. Notice that neither one is anywhere near bulb B. The wires in box 2 are all the wrong wires. Literally every wire except the two you need. So you need to run a cable from bulb A, because that's where switched-hot and neutral coexist.
If you want all 3 to work as multiple control points, then you need to fit a "4-way" switch for switch 2. A four-way takes two messengers on the left, and two on the right, and either passes them straight through, or swaps them. You would feed always-hot at switch 1 instead of 2.
Or junk all this, and go Insteon or other smart switches.
To start with you can only have two three way switches in any lighting circuit, and it looks like you are trying to switch a neutral from switch "1" to light "B". So here are a few basic diagrams one with just two 3-ways and one with an added 4-way.
I set this up so you can turn all of the lights on and off at the same time. If you want to know how to turn the lights on individually. You will need to indicate which switch controls which light.
EDIT FOR CHASE