I was working for UPS since December 2010. I have been paying union dues for over 2 years, and have never had any issues with them. I had received an intent of termination letter from my supervisor in a meeting in which my union steward and i were the only other people present. I spoke with the steward after the meeting , he told me that we would file on this and that it don't mean "crap" Don't worry, We'll take care of this. A few days passed until the next time I would speak with him. He said we will get that grievance, don't think I forgot about you, I'll get with you. A few more days passed by until I was called into a meeting with my Center manager, supervisor, and steward where I was fired. Union Steward never said a word or looked at me. Can I sue union for breaching contract?
This needs to be looked at carefully by a labor lawyer. The duty which a union owes to a member of the bargaining unit is somewhat different than that owed by, say, a lawyer to a client. The lawyer has a duty of undivided loyalty to the client, but the union represents EVERYBODY in the bargaining unit, not just the individual member who wants to grieve. The union owes a "duty of fair representation" to its member. It cannot refuse arbitrarily or discriminatorily or in bad faith to process its member's grievance, but it CAN refuse to process the grievance for a nonarbitrary, nondiscriminatory reason. So in your circumstance, it's not clear why they didn't process or file your grievance.
Moreover, we don't have your collective bargaining agreement so we don't know whethe the union had to process it or whether you had the right to file the grievance yourself. Typically CBAs reserve to the union, rather than the member, the right to invoke the higher levels of the grievance and arbitration process, including taking a member's grievance to arbitration. The time limits for grieving are typically very short, and a member who misses them loses his or her rights to grieve management's action. So, read the grievance and arbitration provisions in the CBA carefully and comply with them fully by grieving management's actions in a timely way in order to maximize your chances of preserving your rights. And seek out a labor lawyer in Texas right away so that you may be fully informed as to your circumstance and your possible courses of action.
Not legal advice as I don't practice law in Texas. It's just my two cents on the facts you describe in light of general principles of law. If you need legal advice, please consult a lawyer who holds Texas licensure and who practices in the area of labor-management relations. That's not me.