Have a basement with 4 beams that had 2 lights in bay 2 and 2 lights in bay 4. Dropping ceiling and want 3 LED lights in each bay. Can I run 3 new 12/2 wires to bays 1,3,5 which will be spliced to the old 12/2 rerouted from bay 2 and new wire ran to bay 2 in a new junction box?
Here are some samples for how a 3-Way switch is wired.
Depending on how your power comes in via the switch or the lights will determine what to use. Basically the Ceiling Box in the diagram is your junction box and the Light in the circuit is all of your LED's you are running from that Junction box. Remember you can't just put 10 wires together and wrap them and claim it is acceptable by code. Daisy Chain your bay runs for your LED's (all led's in one bay are connected in parallel) - so you have 1 line feeding each bay (4 lines from the LED bays to the junction box [J-Box]. Those would be wire nutted together with the switch / neutral and ground. See the diagrams below. Depending on how your 3 way switches are wired up - will determine what you need to do and how to make it right and easy on yourself.
First Image is a brief look at what you should have when you are done.
This is a 3-Way that depicts the power going through the switches.
There are other ways a 3-way could be wired but I felt that this would be easy enough for you to follow and if you needed to (if yours was wired power through the lights for example) - you could research and find that.