I sold my first story to Seventeen magazine - one of the shrinking number of mass-market magazines that still publishes fiction. No agent. I just printed up my story, wrote a cover letter saying who I was and what I'd done, and mailed it off, and was thrilled and delighted a few months later when I got a phone call….and, eventually, a check.
Now, granted, I went to Fancypants U., and I was able to do some name-dropping on my cover letter. Did that help? Sure, probably it did. Is it necessary? I don't think so. I think if I'd submitted the same short story (it was called "Tour of Duty," and published in the spring of 1992), with a letter that left out all the stuff about Princeton, and just said I was a recent college graduate working as a reporter, the story would have met with the same happy response. No matter where, or whether, you went to college, good writing finds a home.
Not all of it applies to everyone but it's worth a read.