Not necessarily.
Some schools are prestigious largely due to their excellent research output, which may or may not translate to consistently high-quality undergraduate instruction, mentoring, and research opportunities.
In contrast, some small colleges that are not very well known (or prestigious, except among the relative few who know them) do offer consistently smaller classes and good instruction by experienced professors (never grad students and rarely adjuncts).
A small, relatively obscure but high-quality LAC such as Lawrence University or Rhodes may not give your kid a full ride … but it may offer enough merit money to bring the net cost down to in-state public university levels. For a Harvard-competitive upper income (> $200K/y) student, that’s probably the kind of school I’d consider “second tier” (just in terms of selectivity and net cost). For a full ride, you’re more likely to be looking at large directional/regional state schools located in places considered (rightly or wrongly) less desirable.
http://automaticfulltuition.yolasite.com/
http://talk.collegeconfidential.com/rhodes-college-memphis/1072290-rhodes-college-for-premed.html