I believe that a lot of your question depends on the dynamics of each specific company. "Medium-to-large company" helps narrow it quite a bit, but each "large" company is different.
I work for a fairly large company (Fortune 500), and I have close personal friends that are recruiters in the HR department. At our company, however, the recruiters have no leeway at all as far as offering more incentives. If the candidate is deemed very, very desirable, approval from three different tiers of management must be had before those incentives can even be offered (this includes salary, vacation, benefits, etc.)
Also, the larger and more prestigious companies, my company included, generally view most candidates like this: "Take it or leave it, the offer is what it is. If you don't like it, go work somewhere else. We have lots of people that want to work here." As I said, exceptions can be made for extremely desirable candidates, that are looking to work in an area that has a shortage of personnel. But generally, most companies feel that in the current job market, you're lucky to be given an offer at all.
I came to work for my company with 4-5 years of career-level experience, and a fairly prestigious previous employer. I attempted to negotiate for a slightly higher salary (5K more) and was denied. I then tried to negotiate for more vacation and was also denied that. Now, the offer they gave me was by no means unfair or unreasonable. I just wanted to see if I could squeeze anything else out of my situation. My company was very patient with me while I negotiated and weighed my options. But in the end, it was a "take it or leave it" offer.
Now, I of course have no idea how "desirable" you are in the eyes of your potential employers, but my experience is this: If you're going to work for a Fortune 500 company, and they make you an offer that seems pretty comfortable with good benefits and relative pay....chances are you won't get much more out of them.
Just my two cents. If you have any follow-up questions or comments, let me know.
Hope this helped!