Unverifiable makes it hard to believe as well (unless the numbers were reasonable to start with).

But there's an easy check, should they care to do it: call their current manager.

If I'm doing the hiring, I wouldn't expect a candidate to let me talk to their current manager--that's just insane.

If I thought the employee was worth the extra money they were asking for, I'd give it to them. If not, I'd let them walk. Whether their current job is considering matching their offer doesn't really matter to me.

If a candidate asks for a specific salary, I don't care why they want it. I only care whether I think they are worth it, and whether I can afford it.