Is it possible to control 3 light fixtures with 4 switches?
This is possible using standard electrical wiring devices!  You will need (in addition to standard electrician's tools and supplies):


Three three-way switches (switches 2, 3, and 4)
A four-way switch such as a Leviton CSB4-15T or equivalent (linking for the sake of those who have never bumped into one, a good home improvement store should have one kicking around somewhere) -- this will be switch 1
A two gang box for switch 4, with a switch + blank two gang faceplate -- one side of this faceplate has the standard switch slot, the other is blank.
And lots of 14/3 NM-B (in addition to 14/2) because you'll need to run multiple travelers


The wiring diagram is as follows:


Dotted lines are neutrals, the rest of the colors are self-explanatory.  Three things:


You will want to use a two-gang box for Switch 4, as the wire fill (eight wires) is too dense for a single gang.  Perhaps you could stash a wiring diagram in there for the sake of anyone who has to fix it later, too?
Keep your travelers consistent: in the diagram, red is up and black is down, always.
Make sure the documentation for this is kept well, otherwise it will give the next electrician who encounters it a bit of a headache.