It all depends on the employer. I have a few friends in the game industry. Here's my anecdotal analysis.
Friend A is a famous game designer who got his start back in the 1980s developing PC games. He helped form a company that went on to great success in the FPS shooter industry. He has no formal education, but tons of domain knowledge after 20odd years in the business. He knows game design and game development.
Friend B is a graduate of Full Sail, which is a video game college, and he was the valedictorian for his class. He works for KingsIsle Entertainment as a senior designer on Wizard 101. This friend got his job at KingsIsle through my connection with Friend A, who used to work with Tom Hall (the founder of KingsIsle).
Friend C designed levels for Quake and other FPS games, then started writing reviews and guides for small video game websites. The last time we spoke, he worked for Microsoft Game Studios as a producer. Friend C has a degree in English.
All three of these people love video games and work very hard at their craft. There are failures along with the successes, but that's how they learned.
The key is if you want to be a game designer, then start designing games. Levels in other games, Flash games, anything. And network! Meet other people in the games industry. Get your work out there and with time and hard work, things will fall into place.