When I was first hired in the games industry, it was based on the merits of my portfolio. At that time, only one of the companies I applied to gave me a test. I saw it as an unusual hurdle.
This time around, however, it became the norm. Virtually every potential employer gave me one or more tests; I ended up taking over a half-dozen of them. Some tests took me only 2 hours, others took me over 2 days.
So, at this point, I feel adequately informed to offer some advice to those seeking game design work.
1) Don't be offended. If a game design company says you must pass a test (or even several tests), don't act shocked. Don't make the assumption that the company doesn't like you, even if someone else who applied there wasn't given a test. Simply, if the company is one you want to work for, take the test.
2) Be timely. Do not take more than 6 days to complete your test. If you're really interested in the company, be done in less than 3 days. If they ask for the test to be completed in a certain number of hours, finish in under the time limit. If you have schedule conflicts, discuss them with your potential employer so you don't have to rush.
3) Be clear and concise. Companies aren't just testing your game design skills - they want proof of your ability to communicate effectively. You must walk a tightrope with each of your answers. You cannot afford to ramble, yet you must explain your thought processes and math choices.
4) Research. All but one of the tests I took was "open book." If you come across something that you're not sure about, Google is there for you. Cite your sources as needed. Plagiarizing is just as unwelcome here as on any test.
5) Edit your work. When you're done with the test, go do something else, then come back and edit your answers with a fresh mind. You'll at least catch some typos (well, I sure did), and you may come across answers you'll want to rework. If it's a timed test, save a few minutes at the end to give your answers a once-over.
6) Pace yourself. Read through the whole test before you start answering questions, so you have a good sense of what you need to do. Design tests can vary widely in content, though the core of most tests will have you design a game, level, or quest/adventure. In general, if you find yourself spending too much time on one answer, come back to it later. You may find that you have fresh insight after working on other questions.
7) Don't worry. Even if you aren't offered employment, by working through the test, you've learned more about game design and you've become a better game designer.