My dad tried that theory once at the advice of a tire shop, put good tres on the front and lousy ones on the back (front wheel drive car). He put radials on the front, and traction C non-radials on the back, which was pretty extreme admittedly. I borrowed it for a few weeks to drive to work when I was saving for my first new car.

Had a spin-out in the first week from what would should have been a simple panic stop from a mere 35 mph. Dented a rim when I hit the curb, felt bad that I had damaged his car so I bought a new rim and 4 of the best all season radials I could find. $400 for tires in 1982 was a big deal.

A week after that a much more major panic stop happened, came up over a small hill on the freeway at 65+ and traffic is at a dead stop. The car stopped on a dime. If I hadn't replaced all four tires I would be likely been dead or seriously injured, and even worse taken someone with me.

The sad end of the story was my mother totalled the car 6 months later (she was uninjured fortunately). At least she got to drive on really nice tires for those 6 months, and I am still here to talk about it. :-)