The Family DuMont

[Electronic M]

Sounds like you are seeing what is called 'retrace lines'. The set is supposed to turn off the electron gun when it is in between scanning different scan lines or frames (via a 'blanking circuit'), but some sets don't have that circuitry (I can't see that being the case for a Dumont though), and others it is either insufficiently designed not working due to bad components.

BTW if you play a VHS tape or DVD that has macrovision copy protection on it then that will partially disrupt the blanking circuit and place a series of bright dashes on screen.