Bhupathi and Nestor beat Romanian Victor Hanescu and Slovak Martin Klizan with a 6-1, 7-6 (8), while Bopanna and Ram lost to Italians Simone Bolelli and Fabio Fagnini with 2-6, 6-7 (3).
Bhupathi and Nestor will now play Bolelli and Fagnini.
Bhupathi and Nestor had a tough time in the second set after taking the first in a jiffy.
The second set saw the Romanian-Slovak pair held their serve to force a tie-break, though they were stretched to save three breakpoints in the ninth game.
Bhupathi and Nestor served well to concede only break point and they saved even that.
In contrast Hanescu and Klizan looked vulnerable serving, though Bhupathi and Nestor could only convert two of the seven break points they forced.
In the tie-break, Hanescu and Klizan gained a mini break to go 4-2 up and had three set points at 6-3.
The Indo-Canadian pair saved not only them but another at 7-8 before snatching three points to take the set and match straight against opponents who concentrate more on their singles.
Klizan is ranked 30 on the ATP charts, while Hanescu is 63.
In one of the early matches to go on court, Bopanna and Ram, seeded 12th, were down a set even before they could settle down, both dropping their first service games to go 0-4 down.
Both good servers, they just could not get into a rhythm and the unseeded Italians seized the opportunity quickly to stamp their authority.
In the second set, Bopanna and Ram managed to force a tie-break, but were soon down 2-5, and could get only one more point before the Italians served out taking both the points.
The big difference was that the Italians hit 33 winners to Bopanna and Ram who could hit only 18.
Both pairs had problems with their serve, though the Italians could convert three of their six breakpoints whereas the Indians could manage only one of four they had.
Both Bolelli (80) and Fagnini (47) are in the top hundred in the ATP rankings, while the former has a doubles ranking of 278 and the latter 110 whereas Bopanna is ranked 12 in doubles and Ram 44.
Ram is actively pursuing singles and is ranked 110.