I suppose it is to just highlight the distinction between 'nouveau' and 'neuf'.
If you say:
J'ai un nouveau vélo.
It means you have a bike which is new to you but not necessarily brand new, i.e., it could be second-hand.
Hope this helps!
"J'ai acheté un nouveau vélo." This sentence doesn't align well with your other examples; that is to say, is it brand-new, or is it simply new to me? The implication is that I bought a 'new' second-hand bike. Is that what you are trying to convey?
I think the confusion stems from the fact that, in English, if I say "I bought a new bike", it would sound funny if this meant I bought a used bike which was merely new to me. It's not impossible that the sentence would have this meaning, but without other context, I would say that someone was being misleading in using "new" in this way. Hence, "J’ai acheté un vélo neuf" would be a more natural translation of the standard English meaning.
(Edit: it seems that the question might be about the placement of the adjective as well, whether before the noun for nouveau or after for neuf. But it's still odd to say in English "I bought a new bike" in general.)
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