Nee Denny. The 5870 is basically issueless with blackops.
My one mate had a 5870 till recently, and blackops had NO issues whatsoever. He "upgraded" to a 6870 like me and he started getting issues. reverted back to his 5870 without changing drivers or ingame settings, Purely just a card swap, issues gone.
EXACTLY the same with a guy in my clan. Has a ATi 4890, oldish card but works perfect still for Blackops, upgraded to a 6950 and issues from beginning to end, went back to the old 4890, no issues whatsover.