OK, I notice that in the "junction" box, you have matched up all black wires and all white wires. As if you expected that all wires just come naturally color-coded like that. No, that does not happen, unless you do it yourself.
Designing a 3-way+lamp complex is not that hard. Certain wires have to go certain places, and they have conventional color codes. But the wires don't come that way by default, so you must forcibly "take over" the wire colors by remarking.
And by "must" I mean for the sake of doing sure, steady and efficient work when you are a novice. Code doesn't actually require re-marking, unless the wire is white.
Further, the topology of the cables must be "Tree" or "Vine". It cannot ever be "Loop". For instance your drawing shows a 14-2 running from the light to the upstairs switch. Nosirree, can't do it.
The 3-way rules (and conventional colors)
You can take a blank sheet of paper, draw the boxes, draw cable routes connecting them, and then add the wires to this diagram as needed. Draw each one:
- two (2) Travelers YELLOW must connect all switches, in string fashion. 2 wires here. There's no need to distinguish one from the other, so they are both yellow. They go on the brass screws of 3-ways.*
- Always-hot BLACK must connect the power source with one of the 3-way switch "commons".
- Switched-hot RED must connect the other 3-way switch "common" with all of the lamps.
- Neutral WHITE must connect the power source with all of the lamps. Switches installed or wired post-NEC 2011 also need neutral wired to one switch (see dashed line here), that's for smart switches.
So draw all that out. Once you do this, it will become perfectly apparent which cables need how many wires, and how to remark them so they are colred by function. You can also see what went wrong in your junction box.
Most of the time it will work out "nice and neat" in the sense that no cable will need more than three wires.
Sometimes, you get into a jam where you need 4 or 5 wires in a cable. (for instance, if we were trying to deliver switched-hot (RED) and neutral (WHITE) to the upper right lamp via the 3-way switch cable, we'd need 5 wires - whoopsadaisy!) The solution is to rearrange the cable topology so the 3-way switches are a "spur line" - a branch that contains all the switches and only switches. This sometimes means running a /3 alongside a /2, but them's the breaks. I believe your setup came to you that way. The wires up the spur need to all be /3 cable (the first segment to bring neutral to a 3-way, and the other segment for 2 travelers and a common).
Other times you'll get a happy accident where you need black, red, white and by golly, that's what's in the cable!
When you build it for real, use the tape to mark the wires. Believe me, you'll thank me next year when you're trying to troubleshoot it.
Marking rules for white wires.
A wire which didn't start hot can never be re-marked to white. However white can be reused as a hot wire if neutral is not present, but there's a pecking order to it.
- White must be neutral if neutral is in the cable.
- Otherwise, if always-hot (black) is in the cable, white must be always-hot. Often you have a black-white cable and you need black-red, and you can't mark the white to red, you must mark the white to black and the black to red.
- White can never, ever be switched-hot (red).
- If you have 2 travelers and always-hot, white must be always-hot (and must be marked).
- If you have 2 travelers and switched-hot, white must be a traveler (and be marked).