In the biggest upset of the tournament, Robredo demolished a badly out-of-sorts Federer 7-6(3) 6-3 6-4 in the fourth round, marking the first time in a decade the Swiss master has fallen before the quarter-finals at Flushing Meadows.
"I kind of feel like I beat myself, without taking any credit away from Tommy," a dejected Federer told reporters.
"I kind of self‑destructed, which is very disappointing, especially on a quicker court.
"I just couldn't do it. It was a frustrating performance today."
As expected, Nadal safely made it through, recovering after losing a set for the first time in the tournament to beat Philipp Kohlschreiber 6-7(4) 6-4 6-3 6-1 at an electrified Arthur Ashe Stadium.
But his win was overshadowed by Federer's unforeseen loss against a player he had beaten in each of their 10 previous meetings, spoiling what was looming as the first meeting between the pair in the Big Apple.
"Two times we were one point away. This time we were one match away. But it's always the same," Nadal said.
"When I see the draw, I think about my first round. If I win, I think about my second. That's it.
"I don't see the quarter-finals or the fourth round before the first round. You know how tough is every tournament, every match."
Robredo was almost as surprised as everyone else after one of the biggest wins of his career.
"It's amazing," Robredo said. "For me, Roger, for the moment, is the best player of all time.
"And to beat him in a huge stadium like the U.S. Open and in a grand slam... it's like a dream."
There was an omen even before the match began when officials moved it from the center court to the smaller Louis Armstrong Stadium following a five-hour rain delay.
U.S. Open champion for five straight years from 2004, the last time Federer played a match away from Arthur Ashe Stadium was in 2006 but refused to blame the court switch for a performance that will raise fresh speculation whether he is capable of adding to his record 17 grand slams.
"That should not be the issue," Federer said. "That's definitely the last excuse you could find."
Robredo will now face Nadal in the quarters, ensuring there will be at least one Spaniard in the semi-finals. There well could be another after David Ferrer joined his countrymen by beating Serbia's Janko Tipsarevic 7-6(2) 3-6 7-5 7-6(3).
The fourth seed will play Richard Gasquet after the Frenchman saved a match point in the fourth set tiebreak before going on to outlast Canada's Milos Raonic 6-7(4) 7-6(4) 2-6 7-6(9) 7-5 in a slugfest that lasted four hours and 40 minutes.
Raonic served 39 aces and hit 102 winners but paid the price for 80 unforced errors.
Earlier, Daniela Hantuchova advanced to the quarter-finals for the first time in more than a decade.
The 30-year-old Hantuchova had not reached the last eight in New York since she was a teenager in 2002 but made up for lost time with a 6-3 5-7 6-2 victory over American wildcard Alison Riske.
"I guess the best things happen when you don't expect them," said Hantuchova, now ranked 48th in the world.
Hantuchova will have to wait another day to learn who her opponent will be after the fourth round match between Victoria Azarenka and Ana Ivanovic was postponed until Tuesday because of the foul weather.
However, it is already guaranteed that Italy will have a semi-finalist after Flavia Pennetta and Roberta Vinci won their fourth round matches to face each other in a quarter-final.
Vinci beat yet another Italian, Camila Giorgi, 6-4 6-2 in just over an hour, escaping before the U.S. National Tennis Center was drenched by a thunderstorm that sent players and spectators running for cover.
Tournament officials were again forced to cancel dozens of matches, before Pennetta returned to finish off in-form Romanian Simona Halep 6-2 7-6(3).
Pennetta has lost in the quarter-finals on three previous occasions and desperately wants to make it through to the semi-finals for the first time.
She said achieving her goal would come at a heavy price because Vinci is one of her closest friends.
"I think it's going to be just a really tough match for both of us," Pennetta said.
"We know each other really well... 20 years or more, because we live almost in the same place, just 35, 45 kilometers (away)."
REACHING HER PEAK
A late bloomer, Vinci has hit her peak at an age when a lot of professional players begin to slow down.
The 30-year-old won just one singles match at the U.S. Open between 2001 and 2010 but in the past year and a half, she has made it to the fourth round or better in five of the last six majors, reaching a career high ranking of 11.
And she has also won three grand slam doubles titles, to share the world number one ranking with Sara Errani.
"I know that I'm not young, but I'm enjoying playing," Vinci said. "I have a high ranking so I'm happy, and I try to stay focused every single day."
New York's fickle weather has been a major talking point at the U.S. Open for years with each of the last five men's finals spilling into a third week because of rain delays.
For years, U.S. Tennis Association officials balked at the idea of building a roof because of the enormous cost of covering Arthur Ashe Stadium, the largest tennis stadium in the world.
But they have finally relented, announcing two weeks ago that they would commence a massive renovation program, which would include a roof, by 2016 at the earliest.
(By Julian Linden)