You can bid whatever you want. What you have to understand with Bank owned properties is that some don't accept any contingencies at all (mortgage, inspection etc.) if they agree to contingencies than banks will usually want to colose in no longer than 30 days. So you must be prepared to move fast!
You can offer whatever you desire. The Bank will evaluate your offer along with any other they receive. More than likely, they will ask for your "Final and Highest" offer. They, then will determine the offer that is the most secure as far as loan approval and which will net them the most money after expenses. Get your loan approved before you start and have your own Buyer Agent representing you. Come to my Free Home Buyer Seminar November 15th at Strayer University in Woodbridge. Info on my web site.