Bitcoins are like tangible cash in a sense that you can't overdraft. If you want to spend more Bitcoins than you have, you have to get more Bitcoins first. To achieve that either B has to send A some Bitcoins, or A and B have to create a transaction together (A created one input, signs it, B creates another input, signs it, and then someone needs to do some balancing of how much Bitcoins go where).

Alternatively, one can expect to see some Bitcoin banks in the future that allow you to overdraft from their shared address, and later charge you for it. This solution is not hard to implement, but I don't know if something like this is available anywhere currently (maybe through IBB). The main problem with this solution is trust between parties and how effective debt collecting would be.

The only way that this could happen is if someone else was willing to offer a line of credit. I am far from an expert but it strikes me as a pretty high risk loan and therefor will be done at a high interest rate.

I can't imagine any mainstream bank wanting to touch bitcoin at the current time. I would imagine getting past thinks like anti-money laundering regulations would just be way to much of a pain for the amount of volume that would be involved. I used to work at a bank, Anti-money Laundering regulations are things they take VERY seriously. Messing that up is a good way to get hit with million dollar fines.