I can say 100% that yes, you can. The only restriction is that you must have your pokedex. I have tried this, and can confirm. I know the pokedex thing because I attempted to access the bank with a different game in, and that is specifically what Bridget says.
On top of that, the bank is only tied to your Nintendo ID. I have traded between three different games with the bank already, two Ys and an X. So that is 100% possible, as well.
Keep in mind you can't use this to transfer items. If your Pokemon is holding an item, it will be sent back to your bag, and if the bag is full it will be deleted instead.
This was a while ago, so you might've already done stuff? but those pokemon are probably going to be deleted.
Strictly speaking, the NNI isn't the kind of thing you putz around with. It's a dangerous thing to delete out of nowhere.
Why? Because the NNI is connected to the currency in your account. It is what allows you to buy things. And the money doesn't transfer between regions (you can't change dollars to yen) so you'd lose that.
In that link before, there was something said that could be considered pretty scary. Basically (emphasis mine):
Can I register the same Nintendo Network ID on more than one Nintendo 3DS system?
No, this is not possible. Your Nintendo Network ID is tied to your software and/or additional content you have purchased and downloaded. As the rights of usage for software you have purchased apply only to a single Nintendo 3DS family system, you can only register your Nintendo Network ID on one system.
The bolded statement makes people think they cannot ever change their NNI. But that's not the case because you can add a new NNI:
What is retained:
All game licenses purchased prior to linking a Nintendo Network ID.
Funds added prior to linking a Nintendo Network ID. (See "What is NOT retained" for restrictions.)
Additionally, if a Nintendo Network ID is linked to both a Wii U and a Nintendo 3DS, the Nintendo eShop balance is consolidated and shared between them.
All free software demos downloaded prior to linking a Nintendo Network ID.
As far as deleting goes, though... completely different story. you can keep your stuff if you format your system memory. That's because:
Deleting a Nintendo Network ID will also delete the following information associated to the account: Nintendo eShop activity and balance, Friend List, Miiverse activity, and registered e-mail address.
Now! I know what you're thinking: These are prior purchases, they don't have anything to do with my NNI, blah blah blah," but, here's the thing-- deleting your NNI has a process to it, and THIS is a part of the process:
Agree to the following points:
The Nintendo eShop and all associated purchases will be deleted, and cannot be downloaded. This applies to purchases made with this Nintendo Network ID both on the Wii U and Nintendo 3DS.
Any remaining balance in the Nintendo eShop account will be deleted and cannot be recovered or refunded. This applies to funds added to this Nintendo Network ID both on the Wii U and Nintendo 3DS.
The Friends List associated to this Nintendo Network ID will be deleted.
Any posts and comments made on Miiverse will be deleted.
Long story short: If you do it, do it carefully.
Best Answer
Do you have any other Pokemon games? If not, you will not be able to access PokeBank as it REQUIRES a game to be used, otherwise you will not be able to transfer. Any of these games are compatible: