Summer Seminar is NOT a checkbox on your academy application. Summer Seminar has absolutely no impact on whether or not you receive an appointment or not.
Summer seminar is designed specifically to give applicants the opportunity to see if the academy is right for you or not. As far as getting turned down for it, that is not uncommon. Only a few summer seminar sessions can be done, so that means only a few hundred people can attend. The academy tries to get people into summer seminar they think it will help. They try and get people from all over the country to make a nice mixed group.
But again; and most importantly, summer seminar is not a check box on your academy application. It has absolutely 0% impact on whether or not you receive an appointment. The only thing that's a positive for "APPLYING" for summer seminar, is that you can put it on your Senator/Representative's nomination application. It shows that you really want the academy and you applied for summer seminar and every other thing you can. Whether you got accepted to summer seminar doesn't even matter so much by the MOC. They know only a couple hundred can attend. But they like to know you applied. But again; that's just for your MOC nomination application. It means absolutely nothing on your academy application. I don't even comment on my applicant's file on whether or not they applied or attended summer seminar. There's no extra points for it.