I had never heard of an offer being rescinded because of a request for increased comp until this thread and @blossom’s post. I do agree with @BKSquared and @tsbna44 that context matters and that you need to have a sense of what your value is. That said, here are few considerations if one is making a counter-offer.
First, is how you ask. One would always start with a preamble about excitement about the job and company (but that becomes white noise as it is almost expected and hence does not convey a ton of information). Tact and respect are important. For reasons that I will mention below, it is important to tie the request to something external to your will (no one responds well to a battle of wills and studies suggest that people will reject economically rational demands when conceding to them is an admission that they are in effect submitting to your will).
Why might this matter? There are a number of studies on women negotiating for compensation. Both male and female managers tend to think women who ask for more in the same language that men use as “not good team players” and do not make the same attribution to men who ask for more. All three of the approaches above tend to mute that reaction. One can reduce the chances of that attribution by pegging an offer to external standards of fairness like market value for the same role or by tying it to the function they would be expected to perform for the team (e.g., in this role, I will be negotiating on behalf of the our team; I’m sure that you would want me to use the same skills in negotiating my position) or by tying it to external advice especially for a younger person (although I’m a little uncomfortable negotiating here, my mentor (Sr VP at XXX) has advised me that I ought to be asking for an amount tied to the market comp). One could start it with “Is there any flexibility on compensation as my sense is market compensation for this position for a person with my level of experience ranges from X to Y … .” If they say no, one can then ask about a signing bonus and other aspects of compensation. While this advice is important for women, I believe that the same advice is helpful for men as well.
In my experience, if one is careful and has a decent sense of the market and says, “My research suggests that someone like me in this kind of role would be paid between X and Y, and where in that range is dependent upon Z and W. I’d be comfortable in the upper half of the range because … . Does that match with your sense of the market?” In my experience with at least a few such negotiations, the recruiter or agent has almost never responded with a description of the market, but immediately offers more. Admittedly, this was in the tech world, rather than engage on the market, they would just increase the offer (not clear whether you see the same effect in a weaker market).
Second, there is what you ask for. It should be tied to clear external standards. “You should pay me more because I am worth more” does not work. The amount requested should not be wildly out of the ballpark, especially in a weak market.
Third, there are a lot of ways that companies can compensate people. In addition to salary, there are signing bonuses, equity/options (for a startup primarily), flexibility on remote work, developmental training, location, title … . The most important is entering a situation in which you are going to have a good chance of being successful: right staff, right reporting relationships, sufficient budget, etc. (most of which are not easy to do for a first year person).