Well, I think I know why some worked and some didn't. I took one that wasn't working properly (not this one - it was 1mH), unwound it, cracked it into four pieces, glued them back together and rewound it. The results: 1mH, saturated at 150mA (which makes sense considering that the 200uH one saturated around 800mA). After modification, the inductance went down to 32uH (yep, a whole 31-fold loss), and the saturation current went up to 5.8A (39-fold increase).
I bet the "good" ones were actually slightly damaged in shipping!
I'll post pictures of the tester and scope captures over at the other thread in a few minutes.