Not very often in this case.
In the last two years of having to check boarding cards, I have never had somebody reach the aircraft with the wrong boarding card due to a computer glitch. Though sometimes passengers seem to have a small collection of cards in their pockets and hand me the wrong one. Sure enough the correct one appears when I tell them their going no further. In that two years of shorthaul flights with three to four flights a day six days a week (excluding hols and days off) I think thats a fairly reliable system.