Maybe a cultural difference considering I'm in Asia but in roughly 1/3 of the follow up interview I've experienced, the HR / manager mentioned there were other motivated candidates. Most of the times it came after I expressed I was hesitant after being low balled hard in term of income.
Assuming they've actually made an offer, I agree that it sounds like something they're firm on and, probably, while they might indeed be willing to hire you, they're perfectly fine with their fallbacks.