I have seen such notes, but you have to take them with a grain of salt. When a department posts a note like that it's sort of the corporate equivalent of "Don't talk to our managers: submit your resume on this web form right here." So you should still talk to professors (but don't waste their time).
When a professor writes that on his/her personal/research website, it means "I already have grad students coming out of my ears, and I don't have enough grant money to support even half of them." This is useful information, because you (should) pick a grad school based on who you want to work with. You need to look elsewhere if all the professors you want to work with at a particular school are over-subscribed. Some professors also become jaded by the sheer number of unqualified candidates who can't hack it: None of the string theory groups in my school will talk to you until after you've been admitted, gotten good grades for a couple semesters and the passed department's second-year screening exam with a good score.
Let me part with this: I got into grad school (probably) mostly because I sent an email to the professor who ended up being my research advisor. I described what I did in the past (which was sorta-kinda tangentially in the same field), omitted any mention of my (not very good) grades, and asked a non-time-waste-y question about the research group. The thing about grad school (at least in the hard sciences) is that one of the criteria for admission is "doesn't anybody want this candidate in his/her research group??" When you have someone pulling for you on the inside, getting in is a whole lot easier.
When a professor writes that on his/her personal/research website, it means "I already have grad students coming out of my ears, and I don't have enough grant money to support even half of them." This is useful information, because you (should) pick a grad school based on who you want to work with. You need to look elsewhere if all the professors you want to work with at a particular school are over-subscribed. Some professors also become jaded by the sheer number of unqualified candidates who can't hack it: None of the string theory groups in my school will talk to you until after you've been admitted, gotten good grades for a couple semesters and the passed department's second-year screening exam with a good score.
Let me part with this: I got into grad school (probably) mostly because I sent an email to the professor who ended up being my research advisor. I described what I did in the past (which was sorta-kinda tangentially in the same field), omitted any mention of my (not very good) grades, and asked a non-time-waste-y question about the research group. The thing about grad school (at least in the hard sciences) is that one of the criteria for admission is "doesn't anybody want this candidate in his/her research group??" When you have someone pulling for you on the inside, getting in is a whole lot easier.