Of course, I'm also an enchanter and engineer, so I have to store materials and equipment for those professions -- not to mention a vast number of quest items that are collecting dust waiting for me to finish the quest. Non-combat pets, items useful in certain instances, items useful for PvP, Darkmoon Faire tickets... my bank just isn't big enough to hold it all.
There are a couple of solutions. Firstly, I can try culling the trash and actually getting rid of some of my quest items, disenchanting equipment I haven't used in weeks, and choosing which of my non-combat pets is my favourite.
Secondly, as I have multiple characters (including a bank-only character), I can send the majority of my non-soulbound items to an alt, who can hold on to them or return them in the mail for easier access. I always lose track of who has which items; I used to use the BankItems mod, but I haven't found one that tracks mailboxes yet.
The final option is, of course, to get more bagspace. Not easy when all your bags are 16-slotters already; I've recently finished the grind for an Argent Dawn Supply Bag, and I'm also hoping I'll come across an Onyxia Hide Backpack or Panther Hide Sack at some point. The materials for Bottomless Bag are offputting, though -- twelve mooncloth for two bag slots? I think I'll have to manage.