I agree with much of what @user7530 has said, and I'd like to add a couple of points based on my observation of some behavior I have observed on this site. The following should not be seen as a justification if rudeness; there is none. But there are behaviors that also transcend the requirement to be nice, and rather than rudeness, I prefer to call it bluntness.
Many people on this site are, like you and me, professional people who are here because, to them and us, math is so fun and a site like this is such a gift that we are willing to spend what little free time we have answering math questions. Honestly, I can't express in words the joy I get when I am able to put together a solution to a problem like this or this. (Maybe watching the Boston Red Sox win the World Series about compares.) That people might benefit from my enthusiasm is for me an ancillary reward. Really, I am here because I love this stuff and I do not get to do it enough in my professional life.
Because my free time is so precious (in addition to a career, I also have a family and a 43-year-old body that needs exercise), I tend to get blunt - prickly even - when I encounter people who demonstrate contempt for math yet insist on being on this site. Such users waste my precious time. These users fall into three categories:
1) Users who demand free tutoring services, begging for help on each microstep. (Some have the chutzpah to ask for such services within a specified time period!) My bluntness is never a personal insult, but instead is designed to make clear that I no longer wish to be sucked into servitude. If users like that never return again to M.SE, then I think we are better off for it; such users do not like math and should not be here.
2) Users who post little more than Maple/Mathematica output outside of comments. I have dealt with one such user who informed me that, since all math can be done by Maple with perfection, actually thinking about math is stupid and a waste of time. (Not his/her words, but a paraphrase.) Why is this person even here? So, yes, I have not been nice to people like this. Not sorry.
3) Users who take the pointing out of errors in their work as insults to be ignored rather than constructive comments that help improve the site. Such users, who occasionally satiate a need for a rep score boost by posting poorly thought-out "hints," express shocked outrage at any downvotes that may occur because they refuse to acknowledge what's obvious to most other users. In this case, I usually point out the error in a clear but exasperated tone; this is not directed at the user but at those who may stumble across the "solution".
Early on in my time on this site, I got into a tussle with @did over my mistaken justification for a step leading to a correct answer. @did was, as usual, absolutely right and corrected a long-held mistaken assumption on my part. I learned something and am enormously grateful to him. But during the exchange, I got rude and testy and refused to believe I was wrong, even though someone who actually practiced math full time was telling me otherwise. So @did got very blunt with me, which shook me into reality. I hope to do the same with others and have the same effect.
That all said, there are lots more new users and experienced users coming here to learn. They ask good questions and demonstrate that they are listening to feedback. We should be doing everything we can to keep those people here, and, as you correctly point out, the language of patience is the only acceptable kind.