I recently got a 1976 Metalux 4 lamp T12 troffer from my dad's church. (He works for maintenance) Well, after I got it home and cleaned it up I tried it with the original ballasts. The ballasts were 1976 Green Label Universal. I put in 4 new Philips F40/CWS. They both worked good for about 10 minutes, then one of the outside lamps went completely off, and one was dim and striating, as if EOL. So I turned it off and tried it again, only the inner lamps lit up. So then, I replace the offending ballast with a known working one of the same type, and the same thing happens! By this point I start to get irritated, and I turn the fixture on, and -read carefully- I rotate one of the inner lamps, they go dim, and the outside lamps fire up!  So I rotate the inner lamp back, and they are all running. But then, I turn off the fixture, and turn it back on, and now the INNER lamps don't work! WTF!? By this time, I'm REALLY irritated, and I take out both Green Universals, and replace them with Advance Mark II, and now it works just great every time.

Are the ballast wires mixed up? I have that happen to me before.

No, the ballasts were completely original, and had no evidence of tampering. The ballasts were, however running 34 watt for umpteen years.

Was the fixture grounded when you tested it? Next, I would double check the wiring - the spring contacts in the sockets may be loose.

The grounding has no effect once the lamps ignite.
But I will really check the contacts...

I have read somewhere about two lamp tandem operation in Europe. The article stated that sometimes these fixture are mis-wired from the factory, although rare (this applies especially to crossed starter lines). I suppose it could happen here in North America as well. I don't know if such fixtures are wired up by hand or not at the factory, but even machines sometimes commit errors.

I don't think the fixtures are wired by machines...
And if so, the machines make miswiring errors only when someone make such error in their program, well except some wire get loose or so, but not swapping wires. But then the whole lot become miswired in the same way...
Or sometimes it is not miswiring, but swapping the prewired sockets when inserting into the ficxture.

But that doesn't explain why the ballasts operate dependently of each other. When first turned on, only one ballast would ignite the lamps. It was not until one the "on" lamps was disconnected that other set of lamps would operate.

Well, when the fixture was wired by hand, the miswiring on some fixture may happen quite easily. And with the same lamps it may even initially operate both of them seemingly correctly, so the fixture passed the quality check...

I second DetroitTwoStroke - was/is the fixture grounded? Aside from the first time a set of lamps turned off, it sounds like missing or poor grounding, and the lamps start when you provide the grounding by touching them. Other than that, sounds like a lampholder or wire making poor contact.

If it would be just grounding, the lamp may not ignite without touching the reflector or so, but after it ignited, it will just continue to light regardless of what the other lamp does.

So either some bad contact (touch sensitive) or miswiring (dependence between lamps; or swapped pre-wired sockets between two tubes, what is in fact just another form of a miswiring).

I have tried to think about possible ballast failures, but I didn't came with any scenario, at least plausible...

I had something like this happen once when I was working on a troffer at school, except one ballast was an Advance Mark III and the other was a Sola electronic T12 ballast that had just been installed. The Sola's EOL protection would trip unless a lamp was removed from the Advance. That problem was quickly corrected by converting the whole thing to T8 IS. Been working great ever since

Well, I'm not interested in Instant Start T8. When I was growing up, Rapid Start was everywhere, and I always loved it. The fixture has always been grounded, since the first time I tried it.