Does "final offer" really mean final?

[Mike Knowles]

I put an offer in on a SFR REO for $15k. Within a few hours, the bank countered at $35k. I upped my price to $20k right away...3 days later they came back at $30k, final offer.

My understanding of the delay is that the negotiator on the bank side needed approval to go lower. What are the odds this is really their final offer? If I came back at $28k, do you think they would take it? If they declined it, would I still have the option of buying at $30k?

The house has been on the market for almost a year. This is a cash offer.