World number one Simona Halep scraped through at the Indian Wells Masters, while Petra Kvitova and Victoria Azarenka were beaten on Sunday.
Halep, the 2015 champion, was pushed by wildcard Caroline Dolehide before advancing in three sets.
However, Kvitova and Azarenka departed at the hands of Amanda Anisimova and Sloane Stephens respectively.
Angelique Kerber and Karolina Pliskova advanced with hard-fought three-set wins.
HALEP SURVIVES DOLEHIDE TEST
Halep eventually got past 19-year-old Dolehide 1-6 7-6 (7-3) 6-2 in two hours and six minutes to reach the fourth round.
The Romanian was tested early before steadying and, once she took the second-set tie-break, a victory always looked likely.
Halep will next face Wang Qiang, who thrashed French 14th seed Kristina Mladenovic 6-1 6-2.
KVITOVA, AZARENKA BOW OUT
Kvitova, the ninth seed, was comfortably beaten by 16-year-old Anisimova 6-2 6-4 in the third round.
The result brought to an end Kvitova's 14-match winning streak as Anisimova broke five times in her victory.
For the first time in four meetings, US Open champion Stephens beat two-time winner Azarenka thanks to a 6-1 7-5 victory in a second-round clash.
Seeds to fall in the third round included Jelena Ostapenko and Coco Vandeweghe, who went down to Petra Martic and Maria Sakkari respectively.
PLISKOVA, KERBER BATTLE THROUGH
Fifth seed Pliskova sent down 12 aces on her way to a 7-5 5-7 6-3 win against Zhang Shuai.
In a second-round meeting, Kerber overcame Ekaterina Makarova 3-6 6-4 6-2 to make it eight wins in 14 meetings against the Russian.