Bopanna and his Romanian partner Florin Mergea, seeded sixth, went down 4-6 4-6 to third seeds Ivan Dodig of Croatia and Marcelo Melo of Brazil.
The 16th seeded Paes and his Polish partner Marcin Matkowski also lost in straight sets to fifth seeded formidable American pair of Mike and Bob Bryan. The result was 7-6 (14-12) 6-3 in favour of Bryan brothers.
Earlier, in a lopsided contest, Bopanna and Mergea struggled to handle Dodig and Melo's fine service games.
The second set followed a similar script and once again, Bopanna and Mergea failed to break their rivals. Dodig and Melo, however, continued to be dominant with their first serve besides converting one of the three break points they earned to outwit the sixth seeds.
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Later for the 43-year-old Paes, eyeing a record seventh Olympics appearance, it was good first set in which the Indo-Polish pair stretched their much-fancied American duo to the tie-breaker. This was after both pairs traded a break each.
The tie-breaker was also well fought before the Americans clinched the set on the 26th point.
The second set was much easier one as the Bryan brothers converted double service break to romp home 6-3 and clinch a place in the semi-finals.