Can a new save withdraw from the pokemon bank?

I've been considering restarting my save and replaying the story with access to the entire National Pokedex (Minus a few Event-only Pokemon I'm missing) from the start. But I would first like to confirm: Can I withdraw my stored Pokemon from the bank if my game is reset? It seems that the only contributing factor is that I end up signed in with the same Nintendo Network ID, so as long as my network ID remains the same, I can withdraw stored Pokemon regardless of what save-game or cartridge is in my system, correct?

While I'm reasonably confident it works the way I think it does, past experience (My Pokemon Ranch being tied to specific trainer IDs, and something about the dream world being tied to the game's wireless settings) has made me just cautious enough to ask before attempting.

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.

I'm not 100% sure, I'd say like 97%, but I'm gonna go ahead and say that will work. I didn't do it the same way, I put Pokemon from X into the bank and withdrew them on a new save in Y. But it worked so I don't see why your situation wouldn't be the same. But, like I said, I'm not saying that it's a sure fire thing.