That would be the question...if he just signed a relinquishment without having legally acknowledged his paternal rights, it wouldn't be effective. And unless you are married at the time of the child's birth, he is on the birth certificate, you and he both sign a voluntary acknowledgment of paternity, or he has a DNA test done, then he wouldn't be presumed to be the legal father of the child.
Texas Family Code
§ 160.204. PRESUMPTION OF PATERNITY.
(a) A man is presumed to be the father of a child if:
(1) he is married to the mother of the child and the child is born during the marriage;
(2) he is married to the mother of the child and the child is born before the 301st day after the date the marriage is terminated by death, annulment, declaration of invalidity, or divorce;
(3) he married the mother of the child before the birth of the child in apparent compliance with law, even if the attempted marriage is or could be declared invalid, and the child is born during the invalid marriage or before the 301st day after the date the marriage is terminated by death, annulment, declaration of invalidity, or divorce;
(4) he married the mother of the child after the birth of the child in apparent compliance with law, regardless of whether the marriage is or could be declared invalid, he voluntarily asserted his paternity of the child, and:
(A) the assertion is in a record filed with the bureau of vital statistics;
(B) he is voluntarily named as the child 's father on the child 's birth certificate; or
(C) he promised in a record to support the child as his own; or
(5) during the first two years of the child 's life, he continuously resided in the household in which the child resided and he represented to others that the child was his own.
§ 160.607. TIME LIMITATION: CHILD HAVING PRESUMED FATHER.
(a) Except as otherwise provided by Subsection (b), a proceeding brought by a presumed father, the mother, or another individual to adjudicate the parentage of a child having a presumed father shall be commenced not later than the fourth anniversary of the date of the birth of the child.
So since he is not a "presumed father" he wouldn't fall under the statute of limitations in TX and this statute would apply:
§ 160.606. NO TIME LIMITATION: CHILD HAVING NO PRESUMED, ACKNOWLEDGED, OR ADJUDICATED FATHER. A proceeding to adjudicate the parentage of a child having no presumed, acknowledged, or adjudicated father may be commenced at any time, including after the date:
(1) the child becomes an adult; or
(2) an earlier proceeding to adjudicate paternity has been dismissed based on the application of a statute of limitation then in effect.
So the long and short of it is that since he has not been established as the legal father, he has no rights. But there is also no limitation on when he could appear and file a paternity suit to ask for them.