Recently I seem to be getting notices that I've learned words or characters at funny times. For example, I think I just got a notice that I learned a word even though I got it wrong. I also feel that frequently I've got notices that I learned a character after I wrote it correctly, even though I had previously got it wrong just a few hours ago (for example, 6 hours ago).

Could it be that you're getting the "word learned" message for a related part that you studied at the same time, like the reading in a combined reading-definition prompt?

If you got a character wrong 6 hours ago, and you get it right now, the interval can double, and then you'll be past the twelve hour mark--anything with an interval more than twelve hours is counted as learned.

I was under the impression that you had to have remembered an item at least 24 hours ago for it to be considered learned. That was my mistake. For the other situation, it could just be my own misperception. It's hard to double check that I'm interpreting the fleeting messages properly.

The (kind of arbitrary) line for 'learned' is 12 hours, actually. But if the interval was 6 hours ago, and you studied it again, the interval would then be roughly doubled, which would push it over the 12 hour mark. There's lots else going on scheduling wise so what happens with the scheduling is highly dependent on context.