There are a few things that are better than the Best Offer. Good credit score, Cash Money on a bid, and a better Down Payment on a offer. Why? Because the Risk is not as great. Bank want to unload a property and not have it come back again, so as far as Risk is concerned "Less is More"!

There are many reasons a bank would accept a lower offer on a foreclosure. In the current economic climate, cash is king. A large amount of cash made the offer very strong. There may also be additional terms that made the lower offer more desirable. The terms could be; a fast closing date, plus the large amount of cash, and a very high credit score,(over 800)
There are things you can do to make yourself successful in the bidding process. 1. Put your offer in writing; 2. Include everything you can to make your offer strong, for example, offer to wave the engineer inspection. 3. Have your finances in order and ready to go, by having your application and papers into your bank so you are MORE than just pre-approved. You can then emphasize to the foreclosing bank that all you need to close is a contract and appraisal. Use a broker who works with an attorney who specializes in foreclosures.
Good luck on your next endeavor. Foreclosures are not easy to purchase. They take more effort and knowledge than the normal purchase. An experienced broker doesn’t hurt either.

Unfortunately, that's how it works. They usually take offers for a few weeks then they take the best offer. Good luck in your next bid!