Adjusting rear derailleur - shifts to lower gear, but not to higher Well, it will shift to a higher gear, but it takes two clicks of the shifter and it skips over a gear when it does shift. With one click the chain hangs on the current gear by the slimmest if threads. It's quite amazing really.

This is a new derailleur, a Shimano RD-M410 Alivio SGS, on a cog set that predates index shifting. Could that be the problem?

I have tried adjusting the barrel adjuster, but that doesn't seem to have any effect (on this problem).

Also, I put bar-end shifters on the bike, so the cable seems a bit short.

Edit

The cog set has 7 gears, but the shifter has 8.

Thanks.
Perhaps your shifter cable can't slide smoothly. Shift to the tallest gear (smallest cog). Pop the shifter housing out of the brazed-on stops. Slide the housing back and forth and look for stiffness. Put some lube in the housing and slide it back and forth to distribute it. 

Perhaps your derailleur hanger is bent. A bike shop will have a tool to easily check its alignment.

If either of these is a factor, then you will never be able to adjust your shifting properly.