If the page originally contained Excel Web App webparts that were referencing Excel documents that have been removed from SharePoint.
When the user attempts to edit the page the WebParts will be unable to open the .xlsx spreadsheets, but no high-level warning or exception gets generated in the ULS, and no feedback is provided for the user.
The solution is to either edit the webpart to point to a different (existing) document, or to enter Maintenance Mode on the Page Properties and remove the faulting webparts through there. After the webParts are removed edit mode starts working again.