I don't have the CT rules.
But in MegaTraveller, [Player's Manual p16] has the line "However, for each subsequent term in which a supply of anagathics is maintained, the character can remain on that line of the Aging Table instead of advancing one line per term as usual. So, if the character is on the age 34 line, he or she can stay on the age 34 line as long as a constant supply of anagathics is maintained."
It looks like they got tired of immortal characters by the time TNE came around. In that rule set, on p. 33: "After about 60 years of use... ...anagathics begin to turn on their users. The manipulation of the body's growth ability that is used to keep the body constantly renewing itself to maintain youthful cells begins to get out of sync. This is first evidenced by the growth of rather unsightly cysts and tumors, and discoloration of certain body areas where local cell growth is getting out of control."
It goes on from there, progressively worse, physical and psychological problems mounting for most of a page. The last line of the section is "(Three score and ten: it's not just a good idea, it's the *law*.)" Emphasis in original.
MegaTraveller was almost CT v2.0, so you could reasonably take the 'maintain supply and there is no downside' approach of MT. Or, if you want to prevent rampant Dick Clark syndrome among your players, start giving them problems after 60 years of use. If they ever get there, that is, that would be a long campaign...