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It can, but it also can depend on what bonuses you have. The random stat gain can be affecting it because it happens everytime you train. Since breaking in your horse is considered training... I wouldnt doubt that it was affected. However, even on my ranch without that bonus it has happened for me. So, I dont think it is a bug or anything
I've noticed that after a horse loses a certain amount of energy, its endy always goes up by .25; when foals are broken in, their energy decreases by quite a bit, so it's perfectly normal for them to gain in endurance.