Big studios tend to want specialists over jacks of all trades. Also, watch out about Game Design degrees because they tend not to be useful outside the video game industry.
Degrees like Art, Animation, Graphic Design, Music, Computer Science (programming), and Computer Engineering(also programming) tend to be useful in other fields. A lot of big names have admitted that the video game industry lifestyle might not be ideal for everyone, which is why they recommend more general degrees.
I would attempt the programming route first since it's the most technical for obvious reasons. Also, for CS and CE degrees, you have to be good at math and science because they are engineering degrees. I say that it would be a good idea to take a $25-100 art lesson to see if you like art or not before paying tens or hundreds of thousands on a complete education.
Some colleges also let you minor in game design.