It wouldn't hurt too ask. I would recommend you ask if its possible to set up an information interview for any who follow in the field you wish to apply.
Just remember when you ask them, you should be brief, to the point, and respectful.
I also agree with Specops115, if you wish to apply for your desired position, be prepared before you proceed with submitting your resume and any other requirements.
Having an internship at Bungie sounds like a pretty sweet deal. :)
I wish you luck.
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I have asked in the past, but got no response. I know the chances are low that they have many spots, and lower that I got one. I've just never seen any postings about it. It's frustrating when all the "junior" listings I find require 2+ years of experience on shipped games.
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Have you tried Linkedin? I've asked several members through that medium, but no response. I know they're very busy and they may not respond for policy, but it doesn't hurt to ask politely from every few weeks. You have to be persistent sometimes. Regarding job experience, I wouldn't worry so much about years of experiences as much. I would be be more concerned if you have the full or near full set of skills they require. You could also look for community or volunteer events in person where you can expand your network. Is what I'm saying making since? I feel I've gone off topic.