No, a union does not have to pursue every case to arbitration. Unions have a duty to fairly represent their members (called a "duty of fair representation"), but they do not have to pursue every case. A rule of reason applies. If the union acts reasonably, it may decline to pursue a bargaining unit member’s grievance if to do so would, in the union’s reasoned judgment, be detrimental to the collective well being of the bargaining unit in general.