Newton,
Banks have so much inventory that they can no longer afford to just blow out properties for next to nothing. Just like privately owned property, time on market will be one of the indicators of what is likely to be accepted. New listings are tending to sell for list price or higher with REO's. There are a lot of rehabbers/investors fighting over them now, so you will have a lot of competition. This keeps the prices at or close to, their current value (reflecting work needed).
To be successful at this, you need to have an agent that is sending you listings daily, have your funds in place (with proof for the bank), quickly estimate repair costs, have an agent that can give you an ARV to see what price makes sense, and you need to be ready to move quickly.

If you need any assistance with this or more info, please feel free to contact me.
Dan Ross
danross@c21-accessamerica.com

You are welcome to bid as much as you would like. However, the bank may counter or even reject your offer. Are you working with a Buyer Agent who is experienced in short sales and foreclosures? This would be a great asset to you in your home search as they could help advise you on how much to bid and the buying process.

If you would like to contact me regarding Buyer Representation, I can be reached via the link to my website below.