If the member legitimately owed the dues in the past but did not pay them, then yes--the club can sue for the past dues. There is no law saying that an organization cannot try to collect past dues, just like any person or business can collect on money owed to them in the past. The only limitation is the statute of limitations: that is, how long a plaintiff (the party suing) has to bring a legal action. Once the statute of limitations has run out, it's impossible to enforce the debt, since it's too late to bring a lawsuit. In Lousiana, assuming the dues arise out of some contract or agreement, then the country club would have up to 10 years to take action to collect them (10 year statute of limitions).