I have been affected by this over the past week or so. I've been on holiday, and came back to a house with no fresh food, and no working bank account to pay for any, so I've been eating processed shite out of the freezer like a student for four days, and it's making me feel a bit poorly, not to mention the freezer is starting to look a little bare. I haven't had breakfast for days, and I have about two meals left if…
I've been with the Nationwide for years now, and never had a problem with them. Being a building society they're a bit less profit driven, though I don't know if anything can protect against offshore cost-cutting exercises in this day and age :/
So if a bank the size of RBS/NatWest (comparable to Bank of America, or Chase) falls over and you don't have an account it might not affect you - unless your employer uses that bank to carry out their pay runs, in which case you're not going to get paid. Several newspapers were reporting people whose house purchases fell through because their solicitors were using NatWest as an intermediary.
(I actually bank with Nationwide as well, although their online service is a bit odd. Then again, when is that not true for a bank?)