Except since chest are private by default you will get an error message saying it is already protected.
However this will leave the chest unprotected, which means someone could come along and make it private to themselves. And I believe also break the chest if they want. Not really a problem on this server but, it is something to take note of.
Really what you have to do is, use both
/cremove
then
/cpublic
and you can check the protection with
/cinfo