I use this analogy when dealing with my bank-owned buyers --
"Sharpen your pencil before you get there"
Essentially, banks will only counter if they deem the offers to be close enough -- on my bank listings, it's generally that deals need to be within 5% of each other...but each specific end lender has different criteria for determination.
Keep in mind these homes are already at a severely depressed price. Banks know the market is depreciating continually, so if you REALLY like one, I'd advise you bid NO LESS THAN full price net, or maybe even $1000 or so more.
That way you're really pressing the issue as to WHY the bank wouldn't take a FULL price offer, etc.