I believe that a lot of this is affected by state laws, perception and the vehicle along with any/all paperwork. Then of course there are the issues of extremes left and/or right; one the one side you could have a car that was heavily damaged and currently only has a dash VIN (cowl and radiator support replaced). In this case it is going to take a good explanation along with paperwork, pictures, etc. On the other side you have basically an unmolested car that has everything which is generally considered above reproach. The majority of the population falls between these 2 extremes and can lean one way or another. In most cases for our hobby, the majority of cars will have the dash VIN and at least one of the other 2 (cowl and/or radiator support). If you have at least 2 of the three (and they match) I cannot see where you would have an issue.
On that episode of chasing classic cars, I think the cars were older cars and only had 1 set of numbers so there was a question about the validity of the cars.