Nemanja Gordic had 18 points to lead Bosnia and Herzegovina.
Portugal fought hard and made their opponents sweat until the final minute but came up short for a third straight time in the Additional Qualifying Round.
They will now have to settle for going through qualifying again in a bid to reach EuroBasket 2011 in Lithuania.
Portugal did go in front in this game.
Their last lead came with 5:40 to go in the second quarter at 26-24 after a basket by big man Elvis Evora.
Mujo Tuljkovic replied with a three-pointer for the home side before Gordic followed with his first six straight points of the night.
He buried a three-ball and then added a three-point play for a 33-26 lead.
A Gordic lay-up at the half-time buzzer took his tally to eight and gave the Balkan team a 39-28 advantage.
Carlos Andrade hit a jump shot to reduce the arrears to 46-41 in the third quarter, only for Bosnia and Herzegovina to score 11 of the next 13 points for a 57-43 lead.
Moncho Lopez, the Portugal coach, made sure his players held their concentration and they poured in the first 11 points of the fourth quarter to close the gap to 57-54.
Joao Gomes capped that run with a three-pointer and the home fans prepared for a nervous finish.
Tuljkovic stopped the bleeding with a three-pointer and Gordic followed with a three-point play.
Jose Costa's three-pointer with two minutes remaining trimmed the hosts' lead to 65-61, but Gordic was too good on this night.
He went to the other end and answered with a trey for a 68-61 lead to ice the victory.
Evora and Costa each had 15 points in defeat for Portugal.