When the right person knows too much.
Sherlock Holmes is the classic example of this. He spends a lot of time knowing who dunnit, but keeping the solution back until it's certain (or until Arthur Conan Doyle has written enough pages). If we were in his point of view we would know what was going on in his head, and the solution would be revealed. If the writer keeps crucial information out of the character's head despite them knowing it, then they can expect to run into problems with readers when the solution is finally revealed, as Agatha Christie did in The Murder of Roger Ackroyd.
When the right person is too far away from 'us'
Again, Sherlock Holmes is the example. He's just too darn intelligent, talented, superhumanly gifted for us to engage with him. We are fascinated by his glittering brilliance, but that's because it's at a distance. Instead, Dr Watson becomes our viewpoint character. He's a decent enough chap, but sometimes a bit slow on the uptake. The reader is always one step ahead of him, and several behind Sherlock.
When the action isn't really in the events that are happening
The example of this is The Great Gatsby. Nick Carraway observes Gatsby, Daisy, Tom etc as they dance around each other. He is the onlooker, not an active participant, in their relationships. But he's the one who is changed by the experience, so while he's not that affected by the external events, he's very affected by the internal changes to himself.