There's not a lot of people that attempt to do it, so I can't vouch how often it's successful, but I have read a handful of stories from FTers keeping their Premier open past the AF and around month 19 the applied for a second Premier - it would be unusual if they didn't apply the same to the ThankYou card family. The only case I've read about have two simultaneous Prestiges open is on MMS. In his case, though, he applied for a second Prestige 8 days after his last, so he did not get the signup bonus, though he did get approved.
So yes, it's possible to have two of the same card, but you need to wait the 18 months after opening the first one to get the bonus again.
I'm confused, you seem to be saying opposite things. Have you read about people getting a second sign up bonus with the first card still open, or not?
Citi will often give out cards where they have no intention of awarding a sign up bonus. And up until recently Citi has been willing to give out additional sign up bonuses on AA cards when someone still had an identical type of card open. What I'm looking for is data points about that with TYPs cards. But it may be too soon to have those data points.