Is there a maximum range on arrows?

You can fix the arrow range in your "skyrim.ini" file.

Add: [Actor] fVisibleNavmeshMoveDist=XXXXX.0000

If for some reason you already have "[Actor]" somewhere in there, just add the next part into that section.

What this does it fixes arrows that can't hit anything past a certain range. E.g., When you shoot something from really far away and it never hit, but when you get a little bit close, it hits. I recommend setting the value (XXXXX.0000) to something around your Actor fade distance (~10000).

Also, the arrows by default are aimed slightly above where your crosshair is. You can fix it with these lines:

Under "[Combat]" add:

f1PArrowTiltUpAngle=0.0

f3PArrowTiltUpAngle=0.0

According the the changelog of the Unofficial Skyrim Patch (which resolves the problem for PC users):

Arrows do not cause damage at long range even if they appear to have hit the target. This is due to an internal distance check in the game that is wrong. This issue has now been fixed through the use of an "ini snippet" which is now included with the USKP distribution. (Bug #136)

This video (screenshot below) seems to indicate that there is no maximum range for a bow shot. I'd still be interested in testing it for horizontal range, but for now I'll just assume that I'm bad at aiming (or just haven't tried enough times).