The problem is, with 12 LEDs anyway (I didn't test any other combinations here), I get the same issue with 3 other USB adapters I have around - about 3.7V (these all charge my phone just fine). Surely there has to be something else going on here?
Note: I have the adapters connected to a male USB connector I got off something else (I checked with a multimeter to figure out which terminal is which). Since I'm using this same thing for each of the USB adapters, I suppose something bad with this would screw up the results on everything.
Edit: WTF? I tried another USB connector and now this works perfectly! 12 LEDs in parallel on the 700mA phone adapter, and measuring 5.017V. I'm not even sure how the other connector would've shorted to begin with, but anyway, it looks like I'll be getting rid of that.
Edit: Fyi, I had a switch wired in with the USB cable (and some more wire to extend it), and as it turns out, the actual USB cable was causing the problem. Now that I think about it - maybe that has a resistor in it by design, because it was used for a USB LED decoration from CVS, and I didn't see any resistors with the LED. That makes sense...