Anyone thinking of pursuing an MFA should read this essay very carefully; it is full of important advice. As a writer who did not pursue an MFA (I had never heard of such a thing), I feel I did fine studying on my own, but I know I would have liked to have peers/friends/community. I finally got some by teaching, and I want to mention that although low-residency programs aren't the beautiful "time out of real life" experience full residency programs are, they have one terrific advantage: your mentor will read and remark on at least 150 pp of your work in a semester. And the residencies are intense; writers do bond.
Lastly, I want to reinforce Tom's comments about the obvious importance of reading your mentors' work and all of the work s/he/they mention. My students report their reading, and I'm often surprised by their lack of interest in anything other than a particular contemporary genre. Learning to read and think as a writer is the real payoff for studying writing. The rest is luck and fate and talent and very very hard work. If you have all of that, okay, connections help, too.