It seems to not be related to cookies, as I cleared out my entire cookies, went back to the site, and it was happy to use my code until I logged in again as myself, at which point it took away the discount and when I tried to add it again, it said I had gone over my limit. So yes, deleting your cookies will let you apply the code, but as soon as you log into their site again, it'll take it away before you are allowed to fully check out.
Looks like they have tied the code to some portion of your account on their site. I suspect if you used a different email and purchased that way (possibly need to set up a whole new account around that email), you could probably use the code again. I don't really need any "new" games that badly as I haven't finished half the ones I already have, and not being able to use the code has helped me save some money this past month.
-merscwog