It could be that one socket in each fixture is defective. Seems like an odd coincidence, though.

I'm not an electrician, but bulbs need a circuit to complete in order to turn on. If it were wired wrong, I don't think either light would turn on. Are you sure those switches control those lights? Is it always the same fixture that doesn't work?

Is this a new problem or one that has always existed?

The starting point for troubleshooting is if you have power at each of the non-working bulb sockets in each fixture. You can use a simple tester like this one:

They are available for $5 at any home center or hardware store. With the power touch one probe to the little tab at the bottom of the socket and the other to side of the socket. If the light on the tester lights, you have power.

I've seen sockets where the center button (which should be the hot side) gets pushed down too hard and the next bulb that's inserted doesn't make contact. If you find you have power to the socket with the switch on and still no light, turn the switch off, verify with your tester, then pull up the button a bit with long nose pliers.

Not "improperly wired" - it's a three way switch pair. Trip/open the breaker first!

Good advice, be careful out there folks.

Update:

Thanks for all the help. I'm too afraid to use the multi-meter on a live connection. I have a healthy respect for electricity and I have no training. I just changed the fixtures and they work fine. I did have some brown burnt areas on the old fixtures and the insulation in the old fixtures looked brown. Maybe someone put in the wrong bulbs . But the bonus is that the old fixtures were ugly and the new ones are nicer.

In a fixture with multiple lights, you usually have multiple wires that all have to connect to the incoming line. They can work loose over time.