The double Olympic champion took the first set 21-15 but tired to lose the following two games 21-10, 21-17 in a defeat that likely brought the curtain down on his hallowed Games career.
"I was pretty drained from yesterday's match," Lin told reporters, referring to his semi-final defeat to arch-nemesis Lee Chong Wei of Malaysia on Friday.
"However in the third game of today's match I think my opponent got lucky because there were several shots at the net that I didn't perform very well," he added.
In the last-four match, Lin won the first game 21-15 but lost the second 21-11 before his dreams of a third consecutive Olympic gold medal were shattered with a 22-20 defeat in the third.
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Saturday's third-place match loss to Axelsen was almost certainly the last time that Lin, who turns 33 later this year, would grace the Olympic stage.
Rio was the tattooed Lin's fourth Olympics and he is expected to have retired by the time Tokyo 2020 comes around. He will be 36 when the next Games start.
"I haven't thought about it that much," he said, when asked if Rio would be his last Olympics.
"I just focused myself fully on this Olympic Games. After this one I will rest for a while and then make a decision," added Lin, a five-time world champion.
Axelsen paid tribute to Lin, who he watched first-hand in China at the Olympics eight years ago while only a teenager.
"No, I would have thought that maybe he would have retired before I reached this level," the 22-year-old Dane said when asked if he could ever have imagined beating his badminton hero for a bronze medal.
"This is so special for me and it's really hard to describe in words how I feel," Axelsen added.
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