Electrical – How to wire two separate 3-way circuits from the same power source

Yes,

You need 14-3 or 12-3 going to each switch from the light/power:

Call the switches L(eft) and R(ight):

Connections@Light:

1. LineInPwr(white) -> Lamp(white)
2. LineInPwr(black) -> L(white tagged black)
3. LineInPwr(gnd) -> tied in common Both L(gnd) and R(gnd)
4. L(black) -> R(black) [traveler]
5. L(red) -> R(red) [traveler]
6. R(white tagged black) -> Lamp(black)

Connections at L switch:

1. L(White tagged black) -> L SW (common)
2. L(Black) -> L SW (brass) [traveler]
3. L(Red) -> L SW (brass) [traveler]
4. L(gnd) -> L SW (gnd)

Connections at R switch:

1. R(White tagged black) -> R SW (common)
2. R(Black) -> R SW (brass) [traveler]
3. R(Red) -> R SW (brass) [traveler]
4. R(gnd) -> R SW (gnd)

Our own BMitch provided an answer and diagrams here (originally found here). We differ slightly in wire assignments (mine's better ;-)(why? both switches have the same wire assignments, the magic happens at the light)), but its basically the same.

Your problem is that the run from the 3 switch box to the top of the stairs doesn't have enough conductors. In addition to the hot and two switched you now have, you will need a neutral return for the new single pole section.

You could use conduit here, but there's no sense ripping out the 14/3 that's already installed. Just run a 14/2 parallel to it as shown.

I'm guessing about which switches must control which lights, but here is a suggested wiring diagram:

B = black, W = white, R = red, where indicated. Wires not marked in the diagram are all switched, so use whatever conductors are available, and please mark the ends of them with a bit of yellow tape for future maintainers. And don't forget to hook up the equipment grounding conductors!

I have omitted the details of the lights wiring as you have already done these.

Pay special attention to the connections in section 1 as the neutral wire here is carrying the load for the entire basement.

Late Edit: I just noticed that in an earlier diagram I recommended what is a code violation in most places. Justin W. has long since finished or abandoned this project, but I don't want to mislead anyone else who might read this later.

I have modified the diagram to show which conductors should be parts of a cable, in order to satisfy modern code, balance the current flow in each cable, and avoid inductive heating, magnetic radiation, and excessive line losses.