I would estimate that 97% of all the apartments we bid for on the behalf of our clients end with a call for best and final bids. For lack of a more eloquent way to put this; it just sucks (if you are a buyer). There is no magic formula for success, but certainly being all cash or non-contingent on financing increases your chances for a successful outcome by at least 60-75% in my humble opinion. If you are going in with a contingent offer, you will have to beef it up a bit if you are made aware that there are cash/non-contingent offers on the table (we must trust the listing agent for this information). So some more speculation on my part; you need to beat a cash offer/non-contingent offer by about 3-5% to have a reasonable chance of success if you are contingent on financing. This is what it will take to get the sellers attention away from the other offers.
One other thought. When presenting your best and final offer, really make it your best, you may not have another opportunity to bid. Unfortunately best and final is not always final, we have witnessed some crazy scenarios after the fact. Yes, some sellers will kick an accepted "winner" when a disgruntled 'loser" re-bids with a big number. As one seller said to me "I may lose a weeks sleep over this (telling a buyer who's offer he accepted after a call for B and F) that they are now out after accepting a much higher bid. That said, many sellers will stay the course with the bid they accepted as a matter of principle.
It remains a difficult market. Just stay persistent and stick to your guns, we are getting deals to contract at reasonable valuations.