I will attempt to answer your question this way:
Within the videogame industry, it is more difficult to secure a job as a game designer than it is to be hired as an artist in a position like character artist, environment artist (which recently I've seen a lot more vacancies for) or similar role. Generally this is because those with the necessary talent/education can get a job as an artist for a game company with minimal professional experience. They will higher junior artists to work under the supervision of senior department leads. Game designers, level designers, and similar roles, are generally reserved for those with several years of professional experience within game development, or, for exceptionally talented individuals with an impressive educational background (usually hired by a small company).
So, there are ample opportunities to go either route, but if you are talking about entry-level work, then you will have to gain experience in a junior level art/animation/programming position before given consideration as a designer, because designers are more senior-level developers.
Now, the specific job of a concept artist for a game company is hard to find, and harder to get. The industry in general is very competitive, but that role in particular is very difficult to get. Usually this is because the creative director and art director handpick this person based on their particular style as it reflects the company's own vision.
For more information about the different art/design careers in the game industry, this site offers an educational resource of career profiles that describe the duties, skills/education, and what to expect for dozens of game careers: http://getinmedia.com/industry/games
Specifically, here is a profile for a game designer: http://getinmedia.com/careers/game-designer