^
My main concern would have been the agent's expertise and/or willingness to spend extra time getting it done at JFK.
I've interlined across two PNRs, one major alliance and one non-alliance interline partner. While I was successful and the agent was willing, she had to make a phone call to figure out *how* to do it. It took about 15 minutes to get it done and tagged correctly.
I can see 3 PNR's causing someone to not want to try and make up a rule about how it's not possible. Star-to-Star is a no brainer, but it wouldn't have surprised me if the JFK agent said "You'll need to get bags in BKK."
Glad it all worked out as desired.