The term "zombie bank" has been increasingly used--a bank with zero value, but kept afloat due to government backing. The best way to gauge this is to look at the financials (below). Look at Citibank & WAMU (before it was taken over by Chase). This NPR story on Zombies highlights the problems. In Japan, which had had a stagnant economy for almost 20 years now (a high wage nation in the aftermath of an asset bubble with tight credit...sound familiar?) suffered from the effects of zombie banks with their recession. There are no easy solutions. Letting zombies fail will just put bad paper (debts/mortgages) in the hands of the strong at a firesale price. In other words, the rich will get richer & competition will go down, increasing industry concentration (oligopoly).
As a follow-up to an excellent question, the chart below shows B of A market share (high). I pulled it from Mergent through the Library portal. The graph doesn't seem to be allowing for an enlarged view, but here's a link to it Flickr.