Part of that is true.
When you upgrade to larger rims its critical that you maintain overall tire diameter. Right now you have 225/60R16'* so if you want to upgrade to 20" rims, you need to find a tire that is very near (within 5-10%) of the stock tire diameter.
Your transmission is geared and operates on a specific rolling distance or total circumference of the tire. If you greatly increase the total circumference of the tire then your tranny will be doing extra work in order to operate properly. The speedo is read off the computer and off the tranny yes. So if your tranny working off a different total rolling distance the speedo will read wrong.
GM corrected this issue with installing new software capability to the 2000+ cars, their computers are Tech 2 and can be programmed to suit new rolling diameters.
I didnt make this up, this came from a GM service manager when I asked about going +1 rims on my car.
Diamond White Classic Aurora - Back on the Road
Galaxy Silver Grand Prix GTP CompG - Soon to be mothballed