I can only speak for start-ups (5-150 people), but here's some general advice given the context of how the business is likely managing the process.

After internal stakeholders have agreed that new hires are needed, a salary range is agreed upon verbally or in email by the CEO/CFO and the person asking for the new hire (lets call them the hiring manager). Experienced hiring managers will probe for things they can offer that are non-cash compensation, like stock and benefits. Accounting/finance is then roped in to make sure everything being offered is sane, and we're just hoping they don't scare the CEO by telling them we're 25% above plan for new hires (this happens a lot).

You end up with something like "Front-end Programmer $120-140k", and finance updates their models to include the new hires so that execs can see the impact to cash burn. Chances are high that many other roles are open at the same time, because doing this ad-hoc is a PITA.

This can take a few days to a few weeks depending on the maturity of the team, and the end-result is some job requirements that get posted on the company website and used to push out to recruiters or personal networks. At this point the CEO is already updating org charts and thinking about how to communicate the new hires in the next board meeting deck, and how to market the work the new hires will be contributing to.

Why is this all important? Because this is the weight of the work you're pushing against when your salary negotiation goes above the high-end of the range.

You see, if your salary is within range then no extra work is needed by the hiring manager or company to make things work. Better yet, if you come in at the lower-end of the range then the company has saved money! Big win, since it always feel like cash burn is faster than expected. Maybe the "extra" cash is added to the remaining new hires pool, or better yet returned to the company for other uses

But if you're above the high-end, the hiring manager has to go back to the business and "fight" to get approval. The CEO has already approved the updated financial model, which the team spent a week or more updating with furious back-and-forth and long nights. So when the hiring manager broaches the subject of needing an extra $X to make this new hire work, a dance begins with internal stakeholders that have already spent weeks on already-agreed-upon numbers. Going through the process of updating all of this can seem daunting.

An inexperienced hiring manager may not know how to tackle this situation, so rather than deal with defending the extra money will just pass on the candidate. Maybe the candidate is a close friend or former colleague, so even an inexperienced hiring manager will figure out how to make a "plea" even if communicated poorly.

Experienced hiring managers will have a few tricks, such as communicating the benefits of the extra cost in a way that appeals to management. Really context-dependent. I've even seen some offers doubled, because the candidate was so appealing and the hiring manager so good at marketing their contributions.

Now here is the hard part - how do you, the candidate, know what the upper range is so that you can ask for it without telling them your prior salary which may anchor you to the lower-end? This is why you want the company to make the first offer, because then you know a bit more about the range. They offer $120k, so you know you can probably safely go to $130k without much trouble and $140k with a little effort. $150k? Well, now the hiring manager needs to begin to negotiate.

There may also be a recruiter in the picture, which doesn't change the internal negotiation dynamics but does make the experience "lighter" for the hiring manager and candidate because there's less face-to-face emotional turmoil.

My advice? Keep in mind the importance of the range, and that you're asking the hiring manager AND other people in the company do more work to accommodate your ask. Don't expect your hiring manager to know how to navigate this process, and be prepared to help them make the case that you're worth it.