Is there a statue of limitations for a broken apartment lease in Tx?

Anyone with access to a crayon can file a lawsuit over anything. The more useful question is whether such a lawsuit would have merit or, instead, be subject to simple and devastating defenses that would result in its dismissal. A suit claiming that you breached a lease agreement would be among the sorts of claims governed by section 16.004 of the Texas Civil Practice & Remedies Code -- specifically, it would be a suit for "debt" within the meaning of section 16.004(a)(3). Accordingly, such a lawsuit must be brought "not later than four years after the day the cause of action accrue[d]." Thus, you would have an affirmative defense to any such lawsuit based on section 16.004(a)(3) -- a defense that would almost certainly result in the prompt dismissal with prejudice of any such lawsuit on grounds that it is time-barred by the statute of limitations.