TL;DR SORRY FOR LONG POST I'm very passionate about this subject--my basic PoV is "Don't bother majoring in art if you don't view it as worth the risk because majoring in art at a non-artcentric institution is generally a waste of time/money. If you can't see yourself as being successful in the art field, major in something else that will make you money instead and MINOR in art."
I have an alternate opinion on this topic and I thought initially about not posting in this thread about it because I didn't want to step on any toes, but I want to offer a counter point of view--I have absolutely no intention of disrespecting anyone's choices or PoVs because obviously everyone's story, life and preferences are different. I just wanna clarify that right off the bat!
I have attended both a public university/state university's art program as well as private art programs and so I'm going to give my perspective based on this.
Going to a public university is not going to make you any more "well-rounded" than going to an art-centric institution, and in fact I'd argue that it'll do the opposite. My first college was the University of New Mexico and I was in the IFDM (Interdisciplinary Film and Digital Media) major, which was essentially the catch-all major for entertainment design-esque fields. I'll be honest with you, it was a hot mess--there were a grand total of 2 animation courses (both 3D-emphasized), a handful of live-action film courses, one course in game design (using GameMaker of all things), random graphic design classes, etc. It was a hodgepodge of different artistic mediums and subjects, which can definitely be a fun thing to explore and introduce you to things you've never tried before! However, if you're looking to make art a career, this would be a severely bad decision to follow through with--ever heard the phrase "Jack of all trades, master of none"?
I feel the need to clarify that even private art-focused institutions require you to take core classes that are not necessarily art related. At the Laguna College of Art and Design I took an ocean sciences class (I already had most of my core classes out of the way from my public uni) but I knew freshman friends who were taking math courses, English courses, etc. However, the difference between the animation program I attended at LCAD vs the IFDM major at UNM was that my non-core classes were tailored and specified to the career path I was trying to take--all of my classes outside of the core credits were directly related to animation. I had to take an animation class Every Single Semester, sometimes two per semester (one for traditional animation and one for 3D!) each semester requiring more advanced techniques, longer scenes, shorter time crunches. Layout, BG painting, storyboarding, character design, etc.--there were no classes that were outside of what you could be expected to do in the career path of animation. Compared to the education I would have received at UNM, I definitely felt like I was being prepared for the actual animation industry and saw clearly where I'd end up in the studio pipeline.
A lot of people here in the thread are vocalizing the concern that art-centric/private art institutions are extremely expensive and are afraid of the debt that can be incurred. To be honest, this is a very valid concern and even with my opinion on attending private art schools being as firm as it is, it's something I worry about myself as well. The thing I think a lot of people don't consider though is that a lot of people at public universities are still required to pay out of pocket or get loans through the government to cover costs--that's a debt you're going to have to pay back as well!
So you're going into debt either way, and the issue is that having a generalized art degree, or specifically a fine-arts oriented art degree from a public university, is very, very, very very very very very unlikely to get you a job at a legitimate animation/game/whatever studio. Extremely unlikely. There are people who DO pay the tens of thousands of dollars to go to the private art schools who still can't find work--realistically, why would a company hire the subpar portfolio with the accreditation from the University of Idaho when they have top-of-the-line applicants from big name schools like CalArts knocking on their door as well? Not only do these CalArts students have the prestige of the degrees as an advantage, but they very likely have the advantage of physical connections--teachers who used to work for Disney that can vouch for the quality of their work and character, friends who graduated a year or two before them who know of an opening in the cleanup department, etc. The reality, unfortunately, is that if you aren't directly within the circle of action (re: big name schools), you're more or less at the bottom of the barrel.
Of course, there are exceptions to this rule, and a handful of people do manage to work their way into the big industry with degrees from public universities away from the coasts, but it isn't responsible to base your future career choices on extreme, one in a million fringe cases.
From my personal PoV and experiences, my opinion on going to an art-focused institution vs. a public university's art program is that if you can't get an education at an institution that won't help you specialize and prepare you for a specific industry, don't major in art at all. If you don't feel confident that you can make a career in the arts or that the risk isn't worth it, don't major in art. You're likely going to go into a little debt at a public university anyway, with a degree that will give you virtually no chances of working in the art industry, and a degree that will not help you move into any OTHER industry outside of art. If you can't see yourself as an artist in your adult career, I really encourage you to consider majoring in a career that you can actually make a living off of and pay off the debt you'll incur from school (even if its only a few thousand dollars) and then minor in art instead. That way, you can still embrace your love and passion and explore the arts, but still have a place to fall back on in case you decide or can't make an art career work.
I'd like to clarify once again that I have no intention of shooting anyone down--in the end, your choices are yours and your situation is likely very different from my own. Remember there is nothing wrong with being an art hobbyist. I know way too many people who stuck around in a public university's over-generalized "art" program and they're stuck working minimum wage jobs because their degrees can't get them any work in the actual art industry and they don't have any other skills represented with their degree that give them a clear career path.
Now that I've said a lot of negative stuff, I want to give some words of encouragement--if you ARE serious about art as a career, if you love it with all of your heart and soul and can't imagine doing anything else with your life or your career, GO FOR IT!!! Don't be afraid! The world has a very funny way of making things work out--obviously finances are a concern, but even the biggest most expensive art schools give out pretty generous scholarships to most entering students that lighten the load of the tuition substantially. You're likely still going to go between 40-80k in debt, but you're pretty much guaranteed a 100% higher chance of getting work in the field that you've always dreamed of--and I gotta say, studios pay their employees REALLY well! The required entry-level salary for animators via the Animation Guild is anywhere between 50k and 70k a year.
I am from an extremely low-income family situation, and actually have been legally separated from my birth family since I was 18. I have financially supported myself since I turned 18 and have been at the literal poverty line since then, facing homelessness, financial insecurity, scraping and scratching to make things work. Being an animator has been my dream since I was 10 years old, watching FMA on Adult Swim and imagining my own stories and characters to the opening song's music. For someone like me, the idea of moving out to California all by myself, one of the most expensive places to live on the planet, and going to art schools that are over 40k a year was something that I felt could only ever be a pipe dream--but I worked hard and I made it happen. At LCAD, my hard work ended up getting me a half-ride scholarship (15k a year). I don't know what lies ahead for me at my new school, Art Center, but I'm positive that I'm going to find a way to make things work, and that I'm going to have a successful career in this field. Separating from my family was a risk, moving to California was a risk, leaving LCAD to pursue a higher quality education was a risk, but that's the thing about entertainment art careers--its cut-throat and risky, but you can't succeed and make your way by halfassing it. If you're serious about it and really, truly want it, go all out--life will find a way to open it's doors for you and make it work!