I was told this by someone with my bank, so I'm not testifying to the truth of it, but it makes sense.
When someone uses their card enough they are actually wearing away at the magnetic strip. The readers in the machine have a certain space between them to accommodate the card. When you swipe a card that is more worn and it doesn't make good contact you don't get a good reading. Wrapping it in plastic (or by putting a piece of scotch tape on the other side) increases the thickness and the plastic is still thin enough to read the stripe.
Sounded plausible to me.