There is no situation where you will say a high number and they will move up to meet you if that number was greater than their max budget to begin with.
Yes, and since it's quite common to have companies where their max budget is not acceptable (probably because they are the ones who can't fill their vacancies and thus have recruiters spam everyone all the time), it's important to filter them out by saying the high number before you've wasted many hours on worthless interviews.
Or you can ask for their expected compensation range instead as well.
Interviewee: I want X.
Company: Sorry but that's above our max budget.
Interviewee: What can you offer me?
Company: Y, Y<X that's our max budget.
Interviewee: OK, I can consider it if you promise me a reasonable revision in a year.
2:
Interviewee: Whats your offer?
Company: Z, Z<Y
Interviewee: That's too low. I want X.
Company: You're asking for X when we offered Z. That's too much of a difference and it doesn't sound reasonable to us. We find it insulting. Let's end it here.
The hiring manager is a person representing a company, which is exactly why they would find their downplaying the offer (because they were hoping to increase it somewhat to meet the candidate in the middle, so they had to start low because they want to keep some gap there) embarassing and feel insulted.
They may not be spending out of their pocket but they are spending out of their budget.