m-ttl,
I went to a state school for undergrad and now I'm at the IFA doing an MA in Asian art history. I also got accepted to Penn - so you shouldn't assume that the name of your undergraduate degree is automatically going to get you rejected from more prestigious programs. I personally feel that with a BA from a state school, I wanted to have a better name from my MA program to balance it out a bit for my future pursuits (which may end with an MA, or may continue onto a PhD, I'm not sure yet). It is, however, obviously pointless to attend a school where no one on the faculty is teaching what you are planning to focus on.
My undergraduate path was basically a result of my financial situation - but there are others like me at the IFA who took a leap of faith with loans for two years (in the end it's sort of like a car payment but way more important and valuable…isn't it?). There's nothing better to invest in than your education. And I think in your case, maybe getting an MA at a school with a bit of a name would be good step towards then going for a PhD at a top 20 school. The thing about the top 20 schools is that they do provide a lot of opportunity for you during and after your program, which a smaller institution with fewer museum connections might not be able to offer. Just a thought!
Also - did you look at SOAS? Quite affordable and it's an extremely prestigious program in the field of Asian studies and Asian art, though it doesn't show up on the top lists because it is so specialized.