Seems like this is a popular topic. :)
Let me first start by saying I am not (and have not been) a video game designer. I have been in Software Engineering for 30+ years now, and have seen the game industry grow and change (along with the software industry as a whole). The advice so far has not been bad, but let me try to add some perspective.
Watching the credits on a feature film (or on a video game) shows you the hundreds of people involved in video game production. Not all of those jobs are glamorous or fast paced... a lot can even be considered WORK. :) But it shows the range of talents that are needed in these fields make a game or film today.
Oddly enough, these are the same skills needed for a lot of other fields. I work in telecommunications, and the need for people to store our software, document what it does, test it works correctly (not to mention actually writing the code!) involves a lot of the same skill sets that people use in the gaming field... or the medical software field... or financial software... or a host of others.
A degree involving software will teach about how to do SOFTWARE, and that is pretty much independent of the domain where it is used. Yes, you do need to learn about domains like gaming, or telecom, or medicine, or finances to be effective in each domain. And liking the field of application is nice. But "sorting data is sorting data" regardless if is in a bank.. or in a game.
If this field fits your interest (and it may or may not... a lot of people start a software degree and don't finish) I think you will find opportunities in and beyond the gaming domain. You might be surprised that a good "configuration manager" (which you may or may not have heard of) may be more in demand at a gaming company than a coder. A good degree at any level in software allows you opportunities in a lot of fields... gaming included. :)
Keep exploring! Learn more about what programs are possible for you, and which are best regarded. Best of luck!