Thanks everyone for your responses.

Each floor does have enough wires for the 3-way and 4-way light
switches. I tried to trace the wires again using a multimeter. After
numerous tries, I'm getting a little closer to getting all the light
switches working. Right now all four lights turn on and off on the
base, 1st and 2nd floors. I'm still trying to figure out the last 3-
way switch on the 3rd floor. I did search online for the various 3-
way and 4-way lighting patterns but the wiring gets confusing because
of the 4-ways.

It's difficult to explain the wiring setup I have but basically the
main power is going in on the base floor and flowing to 1st then I
assume it flows to 2nd or through light fixtures then to 3rd and
return back down to the base.

For the 3rd floor light wires, one set of wires were taped already,
I'm guessing this might be the neutral. Then there are three wires
left, I tried to wire them in different combinations to the 3-way
light switch but still can't get the 3rd floor light switch to turn on
or off all the lights. So for now, I'm leaving the 3rd floor light
switch alone since the other three floors are working.

Also the light fixtures have CFL bulbs installed.

Thanks again!