I'm by no means a veggie garden expert, but I did pick my radishes when they were starting to come up out of the ground, just as you described. Some of them ended up being a bit small, but they do taste much better when they're smaller (less woody, not as hot).
I've also had luck with sticking the "unfinished" ones I happen to pull back in the ground. I just re-bury them and water them a bit and they keep right on growing. So if you pull up some tiny ones by mistake, just replant them and wait a bit.
We used to wait until we saw the tops of the radishes come out of the ground, so we got some really big ones. Well, of the red ones, that is. Of course, we also used to like to let them get large and then eat them like candy, so that might have been a personal preference.
We also used to grow the Daikon radishes, and those got to be around 2-3 feet long before we harvested them. Very tasty, too.