Watson, Britain's number one, suffered a straight-sets defeat to Romania's Irina-Camelia Begu while Watson went down in three sets to Sweden's Sofia Arvidsson.
Watson recently rose to a career-high 39th in the world but had no answer against a player 14 places lower in the WTA rankings as she slipped to a 6-2 6-4 loss in one hour 43 minutes.
The 20-year-old from Guernsey proved a match for Begu throughout but failed to get the job done on the big points.
Watson forced six break points in the match but failed to convert any, while from the same number of opportunities Begu broke twice in the first set and once in the second to secure the win.
Meanwhile Robson, the world number 43, appeared poised to make up for the earlier exit of Watson after taking the first set with ease before racing into a 4-1 lead in the second.
But with the winning line in sight Robson's form deserted her, eventually losing the second set on a tie-breaker and capitulating entirely in the third as world number 49 Arvidsson ran out a 2-6 7-6 (7/2) 6-1 winner in two hours and four minutes.