If the home you want to offer on is an owner occupied, no bank involved transaction; the best you can hope for is a backup position. If the first offer was to try and renegotiate due to inspection or other issues, and you have a strong back up, you may be able to slip in front.
If the home is a short sale, you may have a better chance. Banks want to see a strong offer and the seller will want to minimize their exposure to any deficiencies.
As John said, get a good local agent to represent your interests and advise you. Best of luck.