The policy on churning varies from to bank. The vast majority of churning discussion here is about Citi, the one bank (that partners with a popular airline) that allows really heavy churning. But Citi only partners with AA, so that's the only airline that works with. (Citi applications don't typically say one bonus per member, they say one bonus per account, and it appears they mean by credit card account, not by frequent flyer account!)

One poster has reported that churning also works with two banks that currently have cards for US Air (but it's not clear to me whether it's as heavy churning as AA that they allow). But those banks don't have any cards for any other airlines, and US Air seems to be the least popular of the "big 6" legacy airlines based in the USA.

At other bankds, there may be a little churning allowed, but it requires loopholes. For example, some other banks may only allow one bonus per type of card, but have several types of cards and may allow you to get one bonus on each (at the very least, one personal card and one business card, but maybe even one person "Signature" card and one personal "World" card, etc). UA, at best, seems to maybe fall into this category, as far as I can determine, but exactly which cards they consider "different" enough I'm not absolutely clear on. (In my case, I had some fee-based UA card a few years ago, but for the last few years have only had a Bronze UA card. I have no clue which, if any, of the current UA personal cards they would consider "different" enough from what I have or had to give me another bonus.)

Then there may be "very slow" churning, ie, those banks'airlines that let you get another bonus, but only after enough years have gone by and their primary records of your previous bonus have "rolled off" or whatever...