While I'd love to say, "Absolutely!", the truth is, it's tough. Not impossible, but tough.
Investment banks look for the smartest and most hardworking candidates, and, rightly or wrongly, education pedigree is a big part of that screening. The majority of analysts come from top-tier private schools, as well as a handful from top-tier public institutions with good finance programs such as UVA, Michigan, and UNC. Assuming that your school doesn't fall into the categories above, there are a few things you can do to increase your chances of getting in the door.
The first step is obvious: identify the banks you're interested in and apply as anyone else would. Contact (and be especially nice to) their HR/Recruiting departments, and submit your resume for consideration. This is just a baseline measure to get yourself on file, but you may also gain an ally or two in HR along the way. Keep in mind that while the halls of investment banks are wallpapered with Ivy League diplomas, the HR/Recruiting departments are typically more down-to-earth in that regard. And besides, everyone loves an underdog :)
Second, another obvious point: use the HELL out of your alumni network. I don't care if you go to Southeast Podunk State U., SOMEONE from your school has taken the path you're interested in. Identify these people through your alumni directory or career services office and don't be afraid to ask for help. Chances are they were in a similar predicament as you not too long ago, and while they may not ALL come to your aid, the odds are that at least one will offer a hand in getting your resume in the right pile.
Lastly - and this one's a biggie - don't limit yourself to just the big-name banks. It's easy to get sucked into chasing the Goldman Sachs, Morgan Stanleys, and J.P. Morgans of the world, but there are quite a few smaller, boutique banks where you'll get every bit as good (and possibly better) of an experience. The big banks have so many employees that it's easy to become a very small cog in a big machine. Where at a smaller bank, you're less likely to be pigeonholed, more likely to get exposure to company leaders, and far more likely to take on greater responsibility. And once you've learned the ropes, after 2-3 years you'll have hurdled the school biases and become just as qualified as anyone to work at any bank, big or small.
When all is said and done, the hardest part is getting into the industry. But once you can show – even at a smaller bank – that you're every bit as capable as the next guy, you're on (or damn close to) equal footing going forward. Fwiw, during my time in banking, an employee's effectiveness and school reputation were highly UNcorrelated. I met plenty of morons from Harvard and rock stars from Penn State.
