I don't think this is necessarily the case, because second-tier schools may have more money to spend on merit-based scholarships.
When I was an undergrad (1986-1991), MIT and the Ivies had an agreement that they would only offer scholarships on the basis of financial need, because they didn't want to get into an "arms race" offering scholarships to poach one another's best applicants. This deal was the subject of an antitrust lawsuit and while I know MIT settled the suit, I don't know if the policy stands today.
So if you're smart enough to get merit-based scholarships from a second-tier school, and if your definition of "financial need" is not as Spartan as the student financial aid office's definition, then you may still come out ahead by going to a second-tier school.
When I was an undergrad (1986-1991), MIT and the Ivies had an agreement that they would only offer scholarships on the basis of financial need, because they didn't want to get into an "arms race" offering scholarships to poach one another's best applicants. This deal was the subject of an antitrust lawsuit and while I know MIT settled the suit, I don't know if the policy stands today.
So if you're smart enough to get merit-based scholarships from a second-tier school, and if your definition of "financial need" is not as Spartan as the student financial aid office's definition, then you may still come out ahead by going to a second-tier school.