It all depends on what sort of show it won in and what the rules are of any subsequent shows.
I am running a quilt show/contest at a costumer's convention here in Phoenix next May, and I am not restricting entries. I don't care if it's been entered in/won a prize in another show, I just want *my* show to have some great entries!
Other 'serious' shows may restrict prior winners to being 'display only', not competitors.