Disclaimer 1: A quick check on YouTube and IGN has confirmed that other gamers have discovered this glitch independently of me already and found more effective ways of exploiting this than I ever dreamed of.
Disclaimer 2: I bear no responsibility for anything that might happen if you attempt this trick and end up losing weapons and accessories, so you've been warned. :P
Long story short, I believe I've discovered a way to gain theoretically infinite bank storage space in Borderlands 2, but lack the time, patience, and skill to explore how far this can be taken.
In a nutshell, here's how this insanely easy trick works:
1. Max out both your bank space and backpack inventory slots.
2. Ensure that every inventory slot in both is filled.
3. Proceed to "overstuff" your backpack by, for example, completing and handing in a quest where you receive an item as a reward when your backpack is already full.
4. Return to Sanctuary and attempt to exchange (an) item(s) between your "overstuffed" backpack and your bank vault. Instead of exchanging, the selected item(s) will be crammed into your bank vault and will continue to be dumped into it until you hit the official size limit of your backpack, thus increasing the size of its storage space by however many excess items you deposit in it.
Anyone want to experiment? :)