Robson, a wild card entry, let a one-set advantage slip in losing to Belgian Maryna Zanevska 3-6, 6-3, 6-3.
The 2008 Wimbledon girls champion, whose career has hit the rocks after a succession of wrist injuries, had started impressively - continuing the momentum from her three-sets win over Russian Anna Blinkova on Wednesday.
However, Zanevska capitalised on a succession of errors from Robson in the second set and broke twice in the decider to reach the quarter finals.
Robson has been awarded a wildcard for Wimbledon, which starts on Monday, July 3, where she reached the fourth round in 2013.
British No. 9 Jay Clarke produced the shock of the day as he knocked out Ukranian Ilya Marchenko in straight sets to reach the last eight.
Clarke, from Derby, dominated the top 50 player, who played Andy Murray at the Australian Open earlier this year, in windy conditions on Ilkley's Centre Court and took the first set courtesy of a single break.
The 19-year-old saw a match point pass by at 6-5 in the second before saving a set-point in the ensuing tie-break.
Having changed sides at 6-6, Clarke crunched a forehand winner before sealing the biggest win of his career with a simple forehand.