I have googled the meaning of the word "curt", and google provided me with this definition "rudely brief.". When I searched for its meaning on Dictionary.com it seems that it can also be used to mean "short; shortened.". Is the second meaning archaic or can it be used in writing to say, for instance, "The curt mane of a horse."?
Care should be taken in the use of "curt".
While it may not always have a negative meaning by the user, a listener or reader might not always understand. Historically "curt" could be interchanged for "short", but "short" has become the much more common word.
It would be best not to use "curt" except in familiar contexts.