Buying an REO or foreclosure from a bank is usually a bit different than from your standard homeowner. Banks usually have to convene committees and such to decide on offers that have been submitted. In this case, another buyer will probably not "swoop" in and grab the property, but there may be a higher offer submitted in the meantime, which could be considered at the same meeting.
Depending on how desperate a bank is to sell a particular property, it's usually tough to try and force a bank's hand with conditions such as mandatory acceptance dates, "drop dead" dates, and the like. As with all real estate, if you are truly interested in the property, your best bet is to put together the strongest offer possible. If you are looking for a deal, or submitting a low offer, you have to prepare yourself to potentially lose that property.
It sounds like, in this case, you may just have to sit tight and cross your fingers. As always, make sure to consult with your agent/attorney/etc. about any stratgies, and what to prepare for.
All offers may be presented up till the closing date. That's our law here in Illinois. Dealing with a bank can be a stressful situation, hopefully you have a realtor to walk you through the transaction.
Some banks are easy to deal with, some very difficult.
Hope this helps, good luck with your purchase.