Continuity errors appear in productions with substantial crews and dedicated staff watching for exactly that problem. The reason is structural rather than a failure of attention.
The error is created between takes, not within them
Each take is internally consistent, since it was performed continuously. Problems arise only when two takes recorded at different moments are joined, and neither take is wrong on its own.
Those moments can be separated by hours or weeks. A wide shot and its matching close-up are frequently filmed on different days, after the space has been relit and reset.
Nobody watching either take can see the mismatch, because the comparison does not exist until an editor selects one piece of each and places them next to one another.
Coverage multiplies the possible pairings
A scene shot from several angles with multiple takes of each produces a large number of combinations, and continuity would need to hold across every pairing an editor might choose.
Maintaining that across all of them is impractical, so the effort concentrates on elements likely to be visible in the shots most probable to be used.
When the edit takes an unexpected route, perhaps because a performance was better in a take that was not the intended one, it can join two shots nobody expected to be adjacent.
Attention is directed by the frame, not the story
Continuity records concentrate on what is prominent, and small background elements accumulate faster than any log can capture, particularly in busy practical locations.
Anything consumed during a scene is the classic failure, because it changes continuously during a take and must be reset to a specific state before every restart.
Hair, clothing and hands drift similarly. Performers adjust themselves unconsciously between takes, and the change is small enough to pass unnoticed on a monitor.
The edit sometimes chooses the error deliberately
A take with a clear continuity problem but a superior performance will usually win, because viewers respond to performance and only a minority notice the mismatch.
Editors mitigate it by cutting away at the moment of the discrepancy or by reordering shots so the two problematic pieces never touch.
Why the audience often does not see it
Attention during a scene follows faces and dialogue, and the periphery is processed loosely, which is why background errors survive repeated viewing before anyone notices them.
Once an error is pointed out it becomes permanently visible to that viewer, which is the reason continuity mistakes have a reputation out of proportion to how often they affect a first watch.