Just want to reiterate what @TooManyViolinists said:
Also remember that many kids start at a lower tier school and go on to a more prestigious graduate program.
It really does seem that students at top grad schools for music come from all kinds of backgrounds.
It is wise to avoid debt if possible, at the undergrad level. For grad school, one strategy is to do a doctoral program that is funded and get a master’s along the way. Canada, UK and Europe can also be money-savers.
Some BA programs will give merit money for music (or have more generous financial aid). Of course that means 1/4 classes in music, continued lessons and extracurricular performance but if more affordable, can still lead to grad school for music.