There may be two seperate circuits in the 4 lamp assembly.  If one
circuit has a bad bulb or ballast both bulbs can go out.  Swap the inside
bulbs with the outside bulbs to eliminate/determine the bulbs being the
cause.  Swap both inside bulbs for both outside bulbs and see if the
problem follows the bulbs.  If the outside bulbs still don't light I
would guess there is a bad ballast inside the enclosure.  That repair
requires some mechanical and electrical skills.

Andy

There may be 2 ballasts, with one ballast per 2 tubes.  It may be as simple as
one tube burned out (they have filaments too), causing both on the same ballast
to go out.  Look for a tube with a darkened end.  You can swap one out at a time
to test, but replace both anyway if it's just a bad bulb.
The other possibility is a bad ballast.
Other less likely solutions are a wire not seated in a socket..., they just push
in and are held by a barb. With the light turned off, tug on them to see if one
pulls out.  Just push it back in and make sure it catches.
Maybe a broken socket.
Tube not inserted correctly.  The mark or dimple should be down, making the pins
horizontal.
Bad connection on a ballast because not installed by a pro.

Just a wild guess from someone who knows nothing about this:  does it have
2 ballasts and the one controlling the outside tubes has burned out?

Bobble

Fluorescent lamps don't last forever; neither do the starters.  You
probably need new lamps and/or starters.  A compact fluorescent lamp
on the same circuit would not cause your trouble.

Compact fluorescents don't work on dimmer circuits - they either won't
light or they will flicker badly.

Bud