Permalinks resulting in 404 errors over time

Since the upgrade to Word Press 4.0, one of the sites I’m supporting has become unstable. The home page works, but clicking on links to the blog or pages results in a 404 error. This is a permalink problem and switching to default and back to the “Post name” setting fixes the problem…for a while. After a few hours, the 404 errors come back and we have to re-update Permalinks. All the configurations appear to be fine, and as noted, the site works as expected but goes bad over time. Any thoughts?