She won her first international class on Endeavour last month at Olympia, where there was an ingenuous hint of surprise about the enormous thrill that it gave her. "I'd won two competitions at Wembley the year before, but they were only national classes," she said. "The Olympia win felt so completely different, I couldn't take the smile off my face."
Before the greatest of her triumphs, Bradley had met with Kipling's other imposter: disaster. In April 1992, she was leading a horse that was hit by a football which some children were innocently kicking around. The horse lashed out in a moment of panic and Bradley lost her right eye as a result.
Although she found it difficult to see a stride when she first rode into a fence after the accident, Bradley's courage - and her career with horses - survived. She has since learnt how to deal with the different perspective, whether riding a show jumpingcourse or driving a lorry-load of horses. "When I'm going right-handed, I have to turn my head further and quicker than before."
Big offers have been made for Endeavour (known as Ed at home), but his owners, Charles and Dorothy Sibcy, are not interested in selling. They have had so much enjoyment through watching their stallion in Bradley's sympathetic hands that they have bought two new horses for her to ride; another stallion - five-year-old Tinker's Boy - and a gelding called Goldrush Exchange.
Endeavour's rider is not related to the late Caroline Bradley, although she has a sister with the same name. Alison, whose home is in Suffolk, was still jumping ponies when the elder Caroline Bradley, who had been in winning World and European Championship teams, collapsed and died from a heart attack at the Suffolk County Show in 1983. Alison had admired her from afar, but never knew her.
Teachers at Mildenhall Upper School (which she attended with her sisters, Caroline and Jackie) were not impressed by Alison's application. She was a pony-mad child who found schoolwork a bore and she left (without regrets) at the age of 16. She worked for the Olympic rider, Michael Saywell, from 1986 to 1989 before setting up on her own.
The Suffolk Bradleys are a close-knit family. Alison's father (who used to hunt) and her mother (who is happy to clean tack, but prefers to keep a judicious distance from the horses) are still her greatest supporters.
Bradley rode in her first Nations Cup at Linz in Austria in 1993, where she had just half a time fault in the two rounds. Nick Skelton, acting as chef d'equipe, phoned Michael Bullman, the chairman of selectors, to say that Bradley had "a wonderful temperament and should be promoted".
Last July, she jumped a double clear round in the French Nations Cup at La Baule; at the end of August she moved to Skelton's Warwickshire yard, where she is still based. "She's the best jockey around, so I invited her to come and ride my young horses," Skelton said.
The arrangement is mutually beneficial. Skelton has a sympathetic rider to bring on his youngsters; Bradley receives help from one of the world's leading riders. Her own confidence is boosted by being part of a professional set-up.
Endeavour is nine this year, an age when show jumpers are coming into their prime. Though he can be temperamental, Bradley has found the stallion amenable "once you know his little ways." He pulls ferocious faces in his stable, but he has never hurt anyone.
Timed jump-offs are not a problem for the bay horse. "Ed doesn't look particularly fast, but he has a very long stride," Bradley said. "And I always know I can turn short into a fence, because he is such a scopey jumper."
Endeavour is taking a three-month rest at the moment, but he will be back in work in time for the outdoor season. You can expect to see Bradley riding him at some of the major internationals this year; she could even make it to September's European Championships.
Britain needs talented young riders and horses. What a bonus if an attractive young woman can make it to the top, with the help of a butch bay stallion.