The 28-year-old was regarded as the favourite to win gold heading into the Games, along with Britain's Dai Greene.
However, Culson appeared to struggle down the home straight and was forced to settle for a bronze medal.
"It was a very bad run on my behalf," said Culson. "I'm not pleased with it at all. I lost my rhythm towards the end of the race and I couldn't get it back.
"I send my apologies to the people back home but at least I made it to the final."
Culson's bronze was the first medal won at a Games by a Puerto Rican athlete.