@JD7777 Go Brown Bears from a fellow Brunonian! 
The answer to your question is simply yes, especially at the top privates like the Ivies that only offer need-based FA. To take Brown, for example, 45% of the Class of 2024 received need-based FA, which means the remaining 55%'s paying the sticker price of ~75-80K w/loans, scholarships, parental savings; or a combination of all 3.
If cost is an issue, make sure to apply widely (safeties, matches, and reaches.) Arizona schools offer merit scholarships based on a SAT/ACT & GPA matrix, Oregon schools do as well, as do plenty of others. The top-most schools tend to not offer merit aid, as they are able to attract students regardless.