So... I think SparkFun has sold me a bunch of bad level shifter boards.
@ErinRF 's board isn't the first one to have trouble, and it made me wonder if a pattern was forming, so I grabbed one of the kits I'd packed and assembled it myself, and sure enough, it didn't work. Curiously, it did work intermittently when I attached my logic analyzer to it, but that is not exactly a satisfying resolution. The rest of the time, the signal from the SD card changed at times that it shouldn't, and appeared to bounce between low and high. Signs that the culprit is the shifter board. Sigh.
I've made another board, this one using an SD level shifter from Adafruit rather than SparkFun, and sent it off to be fabbed. It will take a couple weeks to get the boards and the stuff necessary to make them into kits, but this TashTwenty Rev 2 board will replace the TashTwenty Rev 1 board. It'll be $17 instead of $15, as the Adafruit level shifter board is more expensive.
If you have a TashTwenty Rev 1 board from me, I'll be in touch to arrange replacing it at my expense, either with a kit for TashTwenty Tiny (immediately) or with a kit for TashTwenty Rev 2 (when the boards come in), whichever you prefer.