Most of the previous answers have been very good ones, I'd just want to add a couple of points.
Once the lender has gone through the BPO process , they tend to be very set on getting that BPO price for the house, meaning if you see " bank approved at xyz" That lender wants xyz. The idea of low balling on a short sale that has completed a BPO is a waste of time. I know it's human nature to try to wheel and deal, but that technique only works on the infomercials.
What a buyer should do when they see bank approved is look at the home and then compare that approved price to other non distressed ( Short sale/Bank owned ) homes in the same neighborhood.
If the approved price is within 10-15 % of the other properties, you are getting some value.
The notion that you can buy a home in a 300,000 neighborhood for 150,000 is pure fantasy
in most cases.
Please contact me if you'd like to discuss this in further detail