At what point do awards become too old to be relevant (for the bio section of your query)?
That's pretty much the answer to everything of course, so the next question is depends on what?
The what is the significance of the award. Win an Oscar? It's never too old to be irrelevant. In publishing, that's the equivalent of a Pulitzer or a National Book Award or the National Book Critics Circle Award. Those are evergreen.
NYT Bestseller is evergreen for #1 of course, and more green for every week you held off the riffraff trying to topple you.
The bigger the prize or accolade, the greener.
Things that are pretty useless to mention, but sure won't hurt you are things like conference prizes, Amazon Breakout novel short list.
What will hurt you is mentioning a faux prize.
Faux prizes are the ones that want big money up front to enter; have so many categories that being in the top three probably means only three books were in the category, and have no prestige what so ever.
These prizes are intended ONLY to make money for the organizers.
The New York Festival of Books is a great example.
The organizers have a LA phone number.
Lots of categories.
They publish a list of winners; I've never heard of any of them.
(Notice the conspicuous absence of PUBLISHER names by the titles?)
Telling me you won this prize is a big red flag. It means you're not only wet behind the ears, you didn't do much research before entering.