Olivo had never won in an ATP main draw until he beat Paul-Henri Mathieu in the first round.
The Bastad champion in 2006 and 2008, Robredo was coming off a fourth-round showing at Wimbledon.
Fifth-seeded Joao Sousa reached his second quarterfinals in three tournaments after beating local wild card Elias Ymer 6-3, 7-5. The Portuguese faces Dusan Lajovic after the Serb advanced when fourth-seeded Jerzy Janowicz of Poland retired with a foot injury while Lajovic led 6-3, 1-1.
Pablo Cuevas of Uruguay beat wild card Christian Lindell of Sweden 4-6, 6-2, 7-6 (2) to advance to his first quarterfinals in more than three years.