Hi Tim, great question!
I myself actually started at a browser game company before going into more traditional console and video game development.
There are quite a few avenues that people could explore if they're looking to break into the industry - some of which I listed below:
1) Language Knowledge - if you happen to speak a second language, this is a great way to get your foot in the door. Especially big companies who have a large international presence are always looking for language proficient people who can either work in customer service or do language specific testing.
2) Show genuine Quality Assurance interest - I've been recruiting for Quality Assurance positions for more than 2 years now and (while this may differ from company to company) - we actually don't necessarily like it when applicants express their intentions to use Quality Assurance as a stepping stone into the gaming industry. Quality Assurance within the industry sort of (for the most part) has an image problem in my opinion - that it is often easily overlooked and summarized as a 'job where you get paid to play games'. And in some of my interviews we will get applicants who apply for a QA Tester position, but actually express their interest to be a game designer or a programmer. There's nothing wrong with showing that you're interested and passionate about game development - but if you're looking to break into the industry through a QA position - I would advise people to make sure they have genuine interest in Quality Assurance, understand that it is a very demanding and important position and make that clear during the interview.
3) Showing pro activeness - Often times when we interview people and look at their CV - we might occasionally either find gaps in their employment history or they will mention a project that they started and then abandoned, in their cover letter. While this won't necessarily trigger red flags - it might be something that a recruiter will focus on and have people elaborate on. We've turned down applicants where it was clear that they had a history of abandoning things and the reason for this again - is that game development (especially with big companies) is obviously a very demanding profession (no matter whether its coding, QA or 3d modelling). It is important for companies to see that you're able to cope with stress and that you can follow through on things.
4) Modding - If you're a modder, then that should give you a good foundation experience for various roles such as coding, scripting, modelling, etc. If you're a modder for a game that was developed by the company that you applied to - then that's even better! However (from my own personal experience) - I would make sure that you also highlight that if you get hired, it is ok for you to drop your modding in your private life if the company deems that necessary. I actually didn't get a job I applied for - because there was an on-going concern by the company that my passion for modding would get in the way of my professional obligations for the company. I don't know how much of a 'thing' this is for companies out there, but if in doubt - I would just clarify your stance and sync up during the interview on where the company stands on modding.
5) Beta Testing - If a company offers a Beta Testing program - make sure to apply for it and become an active member in their Beta Program. For QA positions this is a great way to also get into the mindset of Quality Assurance. We have a running Beta program at my company and often ask applicants whether they are active members. This also shows that you have genuine interest in improving their games even if you're not an active employee of that company (yet) =P
6) Especially for Indie Developers - I would encourage you to obviously showcase your game. Regardless of what position you're applying for - just mentioning in your CV that you've completed a game of your own, and providing a link is a great way to show your interest in game development and your pro activeness.
7) Go out of your way - This ties into being pro active again - but pro activeness is (at least in my opinion) a HUGE important factor in getting where you want to go. I myself never went to study game development. I went into the gaming industry right after high school and 10 years later I'm a QA manager with a team of 18 people under me. And throughout my entire career I expressed my interest to all of my managers to learn as much as I can about game development and always went out of my way to help out wherever I can. So I would encourage people that once you have your foot in the door - go that extra mile! Ask your manager if there's additional tasks that you can do. Though be cautious about sounding like you want to switch departments in order to learn more about a different discipline (unless that is your intention). This might differ again of course - but some managers might get the wrong impression if they sense that you want to switch departments but as long as you're clear about your intentions there shouldn't be any issues. But in summary: go out of your way of what is expected of you and show interest and pro activeness to do additional tasks. If you can demonstrate this from a previous job - then that's even better!
8) Try smaller companies - if you're not having luck with any of the big companies - try to see if there are smaller companies in your area. Choosing companies that do educational, serious or browser games is also a good way to get the foot in the door before moving on to more traditional pc and console companies.
9) Understand that game development is a business. This is a tough realization and sometimes still smacks me across the face. With all of the fun that's attached to this industry - the concept art, the game design, the testing of games - it is important to understand that game development is a business and companies survive on revenue. Some publisher take this to the extreme (in my opinion anyway) while others not so much - but coming to terms with this simple fact will make it easier for you (especially once you move into a more managerial role) to also see things from a corporate view.
These are just some of the ways - of course if you already have friends working in the gaming industry - you could also try having them put in a good word for you and get you setup with an interview! =)