The fifth seeded duo of Bhupathi and Canadian partner Nestor was sent packing 3-6 6-4 3-6 by the unseeded Italian combo of Simone Bolelli and Fabio Fognini in the second round.
Defending champions Leander Paes and Radek Stepanek were shocked in the first round while Bopanna and Rajeev Ram could not go beyond the second.
However, in the mixed doubles, all four Indian players are still alive with Sania and her new American partner Bob advancing with a come-from-behind win.
The third seeded team defeated the US team of Abigail Spears and Scott Lipsky 4-6 6-1 10-4 in the second round.
They will now fight it out with the Czech team of Lucie Hradecka and Frantisek Cermak for a place in the semi-finals.
Bopanna and his Taipei partner Su-Wei Hsieh also moved to the last-eight stage with a 6-1 7-5 win over Daniela Hantuchova of Slovakia and Fabio Fognini of Italy.
Also Read
They next clash with Kveta Peschke of Czech Republic and Marcin Matkowski of Poland.
Sania and Bopanna would fight against each other in the semi-finals if they win their next matches, which means at least one Indian will be in title contention at the first Grand Slam of the year.