Just to take several elements of the story:
1) black bear cubs are born in January typically and stay with their mother 1 1/2 years, so it's plausible to see bear cubs with their mother in August or September.
2) female black bears typically cover a range of 2-6 miles, while males may cover up to 15 miles, but the record excursion is 126 miles (it does not say how far each day they travel):
"Black bears usually don't travel very far. A female bear will often have a range of 2-6 miles, while a male's range may extend over a much larger area (up to 15 miles), and overlap the ranges of several females. Despite their tendency to stay in one area, black bear excursions of up to 126 miles have been recorded."
3) here's a story from three years ago where a Kenyan runner in Maine on an 18 mile run ran to a house and the bears didn't break into the porch where he went. Not sure how far he was chased.
https://www.youtube.com/watch?v=C8rMm1SPEjk
4) Grizzlies can run 2 miles at 25-28 mph, at a rolling lope.
https://www.backpacker.com/stories/ask-a-bear-how-fast-can-you-run
So, it's possible that a bear would follow someone without catching them.
5) several details in the story sound made up or sensationalistic (I had a question as well about the claim that the next morning she ran 12-15 miles back when she said she'd run 3.5 miles away when she encountered the bears and then 5 miles to get away (but always left--maybe she stayed too close to the bear cubs), where the farthest she'd be away from home would be 8.5 miles. The 12-15 miles in the morning would account for her taking longer to get home, where a good deal would be slow until she reached a trail.
6) sometimes bears tree people but do not climb even though they can. I heard stories on to the best of our knowledge from a bear expert who'd survived many times with bears by climbing trees despite being within climbing range for any bear.