In your two examples you have different tenses. He managed to climb the tree. = He has already climbed the tree He could climb the tree = He has the ability to climb the tree but he hasn't yet. You could say "I could/can manage to climb" but there no real past tense of 'can'

I do not think so. No.. He can climb a tree if he can manage to climb a tree. He must be able to manage to climb a tee if he can do it. Those are true; But... Manage = suggests some level of struggle. Can does not suggest struggle even if it exists.

1. "He managed [somehow] to climb the tree." MEANING: It surprised everybody that he could do it, because he is such a weakling. The boy almost gave up though because of the difficulty. 2. "He could climb the tree." MEANING: The boy is years ahead of other children his age. We knew with confidence that he had the ability to do it.

It's not the same. "could" does not mean "successfully completed" could = capability to do something in the past