The Texas non-compete statute reverses the burden of proof in certain limited situations:

If the primary purpose of the agreement to which the
covenant is ancillary is to obligate the promisor to render
personal services, for a term or at will, the promisee has the
burden of establishing that the covenant meets the criteria
specified by Section 15.50 of this code. If the agreement has a
different primary purpose, the promisor has the burden of
establishing that the covenant does not meet those criteria
. For
the purposes of this subsection, the "burden of establishing" a
fact means the burden of persuading the triers of fact that the
existence of the fact is more probable than its nonexistence.

An example is a franchise agreement that prevents a franchisee from competing with a franchiser during the term of agreement or a non-compete that is included within a buy/sell agreement of a business.  In those circumstances the party subject to the non-compete will have to establish that the non-compete at issue is not enforceable.  (For purposes of an injunctive relief, the party seeking to enforce the non-compete will still have to meet its injunction burden.)  In the typical non-compete case this will not be an issue, but always be on the lookout for an opportunity to reverse the burden.