Here's some interesting reading on the subject
Results showing evidence in support of endurance running
The hunt takes place during the hottest time of the day, with maximum temperatures of about 39–42C. Before starting, the hunters drink as much water as they can. Then they run up to the animal, which quickly flees, and track its footprints at a running pace. Meanwhile, the animal will have stopped to rest in the shade. The hunters must find the animal and chase it before it has rested long enough. This process is repeated until the animal is run to exhaustion. The hunts I observed involved three or four hunters starting the hunt, even when some of them were too old or not fit enough to complete it. A team of hunters can track much faster than one individual on his own. In the beginning the fittest runner may adopt an easy pace while the other hunters do most of the work tracking and running. While tracking as fast as possible, hunters are often slowed down when they lose the trail and struggle to find it again. When the others drop out, the fittest runner must pace himself to run down the animal on his own.
You may also want to read the full David R. Carrier article, but it may be behind a paywall.
Research showing results against endurance running
The world’s first archaeological traces from 2.6 million years ago (Ma) at Gona, in Ethiopia, include sharp-edged cutting tools and cut-marked animal bones, which indicate consumption of skeletal muscle by early hominin butchers. From that point, evidence of hominin meat-eating becomes increasingly more common throughout the Pleistocene archaeological record. Thus, the substantive debate about hominin meat-eating now centers on mode(s) of carcass resource acquisition. Two prominent hypotheses suggest, alternatively, (1) that early Homo hunted ungulate prey by running them to physiological failure and then dispatching them, or (2) that early Homo was relegated to passively scavenging carcass residues abandoned by carnivore predators. Various paleontologically testable predictions can be formulated for both hypotheses. Here we test four predictions concerning age-frequency distributions for bovids that contributed carcass remains to the 1.8 Ma. old FLK 22 Zinjanthropus (FLK Zinj, Olduvai Gorge, Tanzania) fauna, which zooarchaeological and taphonomic data indicate was formed predominantly by early Homo. In all but one case, the bovid mortality data from FLK Zinj violate test predictions of the endurance running-hunting and passive scavenging hypotheses. When combined with other taphonomic data, these results falsify both hypotheses, and lead to the hypothesis that early Homo operated successfully as an ambush predator.
Bear in mind that this is targeting the very earliest humans, not necessary those that have evolved the necessary tracking and team working techniques required for endurance hunting.
The paper Why is alpha-actinin-3 deficiency so common in the general population? The evolution of athletic performance., states in the abstract that a nonsense mutation to the gene ACTN3 that improves endurance activities has undergone strong positive selection in human evolutionary history. I'm not sure whether the paper states that the positive selection is as a result of the ability to wear out their food, though.
Declaration of interest: the people who worked on the paper were colleagues of mine.
This article http://www.bbc.co.uk/news/world-africa-24953910 describes how a group of four men in Kenia hunted down two cheetahs who kept killing their goats, over a distance of just four miles. After four miles the animals were so exhausted that they couldn't run anymore and were caught alive. So it's definitely possible.