I've heard this one repeated by women I know a few times, but it sounds like a product of confirmation bias to me. Is there supporting evidence for the claim that if you don't wear a bra you should expect saggier boobs at old age?
According to the Wikipedia article (with sources), not even bra manufacturers claim that bras prevent saggy breasts.
Now, this is of course not conclusive. I have found two studies, that go further and claim that wearing no bra actually trains the chest muscles, making the breasts firmer and thus prevent sagging.
The first is a Japanese study, Breast form changes resulting from a certain brassière from 1990. Their salient finding is that,
But the study only had 11 subjects so its expressive power is very low.
The second study is by Pierrot L., Evolution du sein après l'arrêt du port du soutien gorge, étude préliminaire longitudinale sur 33 sportives volontaires from 2003. Unfortunately, I cannot find this study online. It is referenced by a lot of health sites.
The key points are referred in the linked article (in French) are:
In conclusion, nobody seriously claims that bras prevent saggy breasts (when no bra is worn). Furthermore, two studies tentatively confirm the idea that wearing bras actually causes saggy breasts.
(It should be noted though that this is not an exclusive cause: breasts will start sagging with age regardless of whether a bra is worn, but wearing a bra can accelerate the effect.)