After analyzing marks and gouges on Tyrannosaurus Rex fossils, paleontologists have concluded that T. Rexes ate other T. Rexes. This cannibalism was likely the result of scavenging rather than active hunting, but researchers aren't 100% positive.
The paleontologists note that the T. Rex was the only large, late Cretaceous carnivore in Western North America that would be capable of making these bites. From the PLoS One study "Cannibalism in Tyrannosaurus rex" by Nicholas R. Longrich (Yale), John R. Horner (Montana State), Gregory M. Erickson (Florida State), and Philip J. Currie (University of Alberta):
We argue that these traces result from feeding, rather than intraspecific combat [...] fighting animals would be expected to inflict wounds to the head or vulnerable areas such as the neck and flanks, and not the feet or arms [...T]he absence of healing in any of these specimens is also consistent with the hypothesis that the tooth marks were made on carcasses.
Tyrannosaurus therefore seems to have been an indiscriminate and opportunistic feeder, feeding not only on herbivorous dinosaurs, but also on members of its own species. The traces described here likely result from opportunistic scavenging, and were probably made after most of the flesh and organs had been removed from the carcass. Presumably, an animal feeding on a fresh kill would instead be expected to focus on viscera and large muscle masses, which would provide more food with less effort. For feet, toes, and arms to be an appealing source of food, most of the carcass must already have been defleshed. [...] While we interpret these traces as the results of scavenging, we cannot entirely rule out the possibility that these traces result from an individual slowly consuming a kill over an extended period of time. It does seem improbable that Tyrannosaurus routinely hunted full-grown members of its own species; however, it is possible that intraspecific combat led to casualties, with the dead becoming a convenient source of food for the victors. Still, compelling evidence for predation in Tyrannosaurus remains elusive.
As jarring as the image of one Tyrannosaurus chowing down on another is, this kind of behavior isn't totally uncommon amongst large modern carnivores. Alligators, grizzlies, and komodo dragons are cannibals, and in most of these instances the cannibalism is the result of predation rather than scavenging. Paleontologists are still debating whether the T. Rex was an apex predator or a scavenger, but at least they know it was a consummate baby eater.
[PLoS One via Yale University. Top image from Jurassic Park 3. Yes, we realize that is a T-Rex versus a Spinosaurus, but that movie had no T. Rex vs. T. Rex battle scene.]