That was a bit of a mess. Technically, no carrier at the time had a "real" 4G network, according to the ITU- only LTE-Advanced and WiMAX 2 met their definition.

Since the ITU had declared that Sprint's WiMAX network and Verizon's LTE network weren't real 4G, T-Mobile decided that they could call their (just-as-fast-as-early-LTE-networks) HSPA+ network 4G too, and then AT&T was almost forced to follow suit because T-Mobile was advertising how their 4G network was so much bigger than everyone else's.