In Texas, covenants not to compete are governed by Texas Business & Commerce Code §15.50. I have included a link to the BCC §15 below. You can skim down to §15.50 to read the statute.
Generally, if the non-compete is directly related to the employment, restricted to the employees activities and a reasonable geographical area, and secured through valid consideration (which generally means continued employment or some sort of monetary incentive), it will be enforceable. The most recent precedent from the Texas Supreme Court on this subject is the case of Marsh USA v. Cook. I have included a link to this case below, so you can review it and see that the courts will generally enforce a non-compete agreement which meets the requirements discussed above.
Even if your non-compete agreement is three years old, it may be enforceable if it meets the statutory requirements and conforms to applicable case law. If you are uncertain, you should consult an experienced labor and employment attorney. Good luck.
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The non-compete agreement might still be enforceable even though it was not signed until later in your employment. But it will take a more detailed review of the surrounding facts and circumstances to give you better guidance. A local employment lawyer will be able to help you review it in more detail and determine how best to respond. Feel free to reach out to me if you need further assistance.
This information is provided as a general reference based on the limited information that was provided and should not be considered legal advice. Providing this information does not create an attorney-client relationship.