Heh. I really don't know anything about ABC Family. Do they have a history of killing shows quickly?
Shows in general? Not really - they're no more finicky than other basic cable outlets on the whole.
However, they're a bit quicker killing shows that are really good but off-brand for the network, not fitting into the comfortable boxes of the teen girl demo the network has come to value. Specifically, The Middleman (a really great sci-fi comedy) and Huge (a really sharp teen drama) were two shows critical communities embraced but were eventually too low-rated to survive alongside more "buzzworthy" entries like Pretty Little Liars or The Secret Life of the American Teenager.
Bunheads is at a weird intersection, in that the pilot is very much about Michelle, but the demo of the network is clearly aiming more toward the teen girls who are underdeveloped. Accordingly, some critics (quite rightfully) have questions about how that balance will be struck in the future, acknowledging that they're much more interested in Michelle/Fanny than in the girls, and the network's track record is such that you could imagine the show getting gradually pushed in that direction.
Which, on some level, is just the consequence of the network. Switched at Birth, one of their stronger shows at the moment, always feels like it's compromising something, but the good parts (specifically the use of sign language and the investigation of deafness in a teenage context) are strong enough to make the show worth checking out (it's on Netflix). The question, I suppose, is whether Bunheads can be successful enough among young audiences without comprising the parts of the show that seem so sharp in the pilot.
The jury's out until we see ratings, and until we see next week's episode, but that's out of our hands at this point.