Rodina, a 120th-ranked Russian qualifier, upset Keys 7-5, 5-7, 6-4.
But instead the 38-year-old was worn down by Dutch 20th seed Bertens in a dramatic third round clash lasting two hours and 40 minutes in sweltering heat on Court One.
Of the players seeded in the top 10 ahead of the tournament, only world number one Simona Halep, seventh seed Karolina Pliskova and five-time Wimbledon champion Venus Williams, seeded ninth, are left in the draw after just five days.More news: Samsung phones are sending photos to random contacts without permission
After defending champion Garbine Muguruza suffered a shock exit in the second round on Thursday, it was 10th seed Keys' turn to endure an unexpected 7-5, 5-7, 6-4 third round defeat.
The 17-times Grand Slam champion dug in though and managed to avoid being dragged further into the afternoon, claiming victory when Kukushkin's forehand smacked into the net tape.
Venus had rallied from a set down in her first two matches.
Keys lost nine games on the trot from 5-2 up in the first set to 0-4 down in the second.More news: Rock-solid Brazil living up to billing as World Cup favourites
"Yesterday he was playing so, so good, and hitting the ball so hard that I couldn't do anything", the 82nd-ranked Pella said. She then broke back to level the third set at 4-4, but Rodina immediately earned another break and served out the victory. I thought I did very well today. "I feel ready to compete".
The 36-year-old Swiss extended his winning streak to 29 consecutive sets at the All England Club. Ekaterina Makarova came out on top of Lucie Safarova 4-6, 6-4, 6-1; while Camila Girogi survived Katerina Siniakova to advance to the Round of 16.
The 52nd-ranked Giorgi produced a backhand victor at 4-5, 30-40 in the second set, then won a tiebreaker to force a deciding set.
Nadal will be up against the promising Alex de Minaur, who beat Pierre Hugues Herbert 6-2, 6-7 (8), 7-5, 6-3.More news: Trump administration cuts Obama-era guidelines for campus diversity
German fourth seed Alexander Zverev takes on Taylor Fritz, the world number 68, who is bidding to reach the third round of a Slam for the first time.
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