I am in the post-grad school job search process and am wondering if it is appropriate to apply for more than one position within the same organization? I'm worried that it may show a lack of commitment, but they are all positions I'd be interested in and mostly qualified for.
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I don't think that applying for more than one position within the same organization shows a lack of commitment; in fact, I think it shows just the opposite. That you're interested in multiple positions doesn't necessarily mean that you're not focused on a certain position, but it does suggest how much you'd like to work for the company in question. As long as your qualifications meet the requirements for each position, you shouldn't be in danger of an employer viewing your application as frivolous.
The key is to treat each position as a unique opening with its own specific requirements. The most effective way to prove that you're not just applying for every available job under the sun, and are in fact interested in the jobs for which you're applying, is to tailor each resume for each position's requirements. Tedious? Maybe, but it's important to treat each position as though it is completely unrelated in order to present yourself as a distinctive candidate for each role.
If the company specifically says they don't want you to submit a cover letter, then I'd recommend including each job title in the objective section of your resume. I'm usually anti-objective, but this is one time that it would actually have a sensible function. If the positions fall under different departments and ask that you submit your application to different managers, be sure to include cover letters (if applicable) with each one.