I too was accepted to Cornell, but from a community college recently turned 4 year lol. It is possible from pretty much any kind of school if your overall app is strong. I also never took SAT, and only had a 3.2uw/3.6w in HS (major turnaround my senior year though-3.9uw/4.6w). Just keep your grades up. And you probably already took the SAT, but if not you may need to for the T20 schools. Cornell doesn't require them, but most of them do, I was just fortunate to be accepted to Cornell, and it was my top choice anyways.
To add to what Lurker was saying about recs...what worked for me to get great recs (I am assuming both were, but I read one and it was outstanding) was to go to office hours and just talk with them. I have never really been vocal in class. Just talking with my econ professor and telling him my plans/asking for advice and staying in touch with him over the summer and this past semester got me an amazing rec. With my english prof I was going to his office all the time to edit my college and class essays anyways, so we naturally talked a lot about this stuff and he was really helpful. Now, if you have classes with hundreds of students this could obviously be a problem, but it worked well for me (and I didn't even do this for the purposes of a rec).