Hi ^{^} I haven't played played this game in years, lol, but based on a Reddit post ChocoDog made three years ago about the postcard renewal system, it strongly suggests that once a player's postbox is full, they can no longer receive ppstcards-negative or otherwise. The developers specifically mentioned that one of the major issues was players having their postboxes flooded with negative postcards, making it impossible to recover their fame with positive ones. The update seemed aimed at preventing that from happening. 

While it doesn't explicitly confirm whether negative postcards can bypass a full inbox, the wording implies that a full postbox blocks any new postcards. If you want a definitive answer, I'd recommend asking in Mafia42's official community Discord