Robson - Britain's number one - crumbled on her second serve on her way to a 6-3 6-4 defeat to Daniela Hantuchova of Slovakia after Watson had lost to Russia's Alla Kudryavtseva in little over an hour.
The 21-year-old from Guernsey was playing only her third match since recovering from glandular fever but she rarely threatened a resistance on her way to a 6-4 6-3 loss, and Konta's subsequent 6-4 6-1 dismissal by 12th seed Kristina Mladenovic left Britain without a player in the third-round draw.
World number five Hantuchova took advantage of Robson's faulty second serve to book an appearance against Mladenovic in the next round.
Nineteen-year-old Robson won just seven of 26 points on her second serve, despite winning 92 per cent of points on her first, but her early exit will allow her extra time to train for Wimbledon with new coach Miles Maclagan.
Kudryavtseva is ranked 118 places below Watson in the world rankings but the qualifier raced through the first set before quickly building a 5-2 lead in the second by breaking the Briton three times.
Watson did prevent the Russian from serving out the match but, having only claimed 42 per cent of points on her second serve, she ended the match with a double fault.
Australia-born Konta, meanwhile, struggled throughout against Mladenovic, who had beaten British number four Tara Moore in the first round.
There were wins for third seed Sorana Cirste and fifth seed Sabine Lisicki, who beat Tsvetana Pironkova 6-4 6-4 and Kristyna Pliskova 6-4 6-2 respectively.
Bojana Jovanovski needed three sets to see off the Czech Republic's Andrea Hlavackova 4-6 6-1 6-4 but Croatia's Mirjana Lucic-Baroni had an easier time in a 6-4 6-0 triumph over Yanina Wickmayer.
The only tie-break of the day took place between Maria Sanchez and Alison Van Uytvanck, with the former closing out a 7-6 (7/3) 6-3 win before rain delayed the evening session.