Alfred Hitchcock Presents: The Case of Mr. Pelham (1955)

Mr. Pelham consults a doctor about a series of troubling incidents. Recently a number of his acquaintances have claimed to have seen him in places where he could not have been. The doctor suggests that there must be another man with a strong resemblance to Pelham. But it soon becomes obvious that the other man is also deliberately impersonating him - even to the point of using Pelham's apartment as his own.