If you're not independently motivated enough to strive for the best, work hard in all of your classes, work in a non-competitive but highly intelligent environment, then you should not apply to this school. Yale is amazing, but it takes a lot of hard work that's put forth outside of the classroom environment. On top of that, there are numerous extracurricular activities that require a lot of good time management skills, as well as a willingness to put forth effort to be an active member of whatever organization you choose. Yalies are very busy, very intelligent students.
There is room for almost any kind of person at this school, but to do well, you do need to be hardworking. If you have been accepted, your intelligence will be enough to get you through, as all inequities in educational background are essentially eliminated after the first year. So if you are lazy, or for some reason or another do not like to be challenged (because you will be challenged - by both your professors and your peers), don't come. Also, if you are not particularly tolerant of homosexuals, that may make your time here rather uncomfortable as well.
If someone can't handle not being the best at what they do or are interested in, they shouldn't attend Yale. Yale is a place for people to be inspired by the fact that they are no longer number one at everything, like most of us were when we were in high school, and that there is more to learn. The competition at Yale to be the best will eat someone alive if they can't come to terms with the fact that someone just may be better, and that they should applaud and study it, not envy it.
Someone who is unwilling to become involved outside of class. Students at Yale take their extracurriculars as seriously as their academics, which is pretty seriously. Someone who is looking for a purely academic experience may feel lost in a campus full of theatre and a cappella people, news reporters, athletes. Finding your extracurricular niche parallels choosing a major in some ways.
Anybody who doesn't care, who doesn't really have goals or intend on having them in the near future. They would stick out like a sore thumb. Also, anyone who is generally intolerant of other ways of thinking or being will have a really rough time at Yale.
You shouldn't attend if you're a little closed off because it will be very hard to make friends and meet people which is very difficult to deal with when you're really far away from home and your childhood friends.
Really, Yale is so diverse, academically, culturally, and socially, that I can't say who shouldn't come. It's such a friendly and accepting environment that you immediately feel like you fit in.
If you are a person who does not want to cooperate with others and only wants to get ahead in life for personal gain then you would not enjoy the interactive atmosphere that Yale presents.
Someone who does not want to be engaged with other students, work hard, or be challenged to improve the world around them. In other words, 99.99% of people would have an amazing time.
Someone who is not confident in their abilities and ready to work hard and play hard.