Can I get an overdraft? Hypothetically...

A uses a desktop bitcoin client, who would like an overdraft facility.

B has lots of bitcoins and would like to offer that overdraft and charge interest on the overdrawn amount. 

Can A just go into a negative balance with B's backing? If not, what would A and B need to change to achieve an overdraft or an aproximation of an overdraft? 

(Note, an overdraft is different from a loan, but a loan might be used to approximate an overdraft if its automatic.)
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.