The bank doesn't have to counter. They look at the best and highest offer, and make a decision if the offer is good enough to suit their purpose. And once they accept an offer, they don't even want to look at another offer as back up.
The best offers are those with the fewest, shortest, or no contingencies; substantial initial deposit and downpayment; good financing, or all cash.
Should you make an offer on another foreclosed property, consider that you may only have just one chance at this. Consult your realtor for guidance. Low ball offers are exactly that, and treated as such.
You should also realize that when you find a property that appears to be a good deal, other folks are seeing it too. So yes, we're in a multiple offer situation again ---- and this time, it's on foreclosures!