In the first day of full-draw action, the pair - seeded third and fifth respectively - were each beaten in three sets at the WTA International clay-court event.
Flipkens went down 6-2 6-7 (3-7) 6-3 to the world number 96 Zarina Diyas - despite the Kazakh double-faulting six times over the course of the two-hour match.
Russia's Vesnina too got off to a poor start as she was eliminated by Casey Dellacqua after a 7-5 3-6 6-3 defeat.
Having hauled herself level by the end of the second set, a disastrous early spell in the third, where the Australian moved 4-1 up, helped end Vesnina's hopes.
Eighth seed Alison Riske enjoyed a more positive day however as she proved too strong for Brazil's Teliana Pereira - saving eight of 11 break points to edge the world number 95 6-3 6-3.
France's Claire Feuerstein sprung a surprise by eliminating Canada's Sharon Fichman - despite being separated by 47 places in the rankings.
The wildcard won through 6-2 6-4 and will meet defending champion and compatriot Alize Cornet in the next round while Paula Ormaechea threw away a one-set lead as she bowed out to American Vania King.
King won through 5-7 6-2 6-1 while fellow American Madison Keys is also safely through to the second round.
The 19-year-old battled back from 5-3 down in the first to move ahead on a tie-break against Maria-Teresa Torro-Flor before ending the second in dominant fashion by reeling off the final four games.
Her 7-6 (7-2) 6-2 victory over the Spaniard ensures she meets Riske next.
Ajla Tomljanovic beat Kristina Mladenovic for the second time this year, securing a 6-4 6-0 win to book a second-round meeting with Diyas.