Yes, eventually there is a point that a tire won't hold a tube, but more than likely that point will be discovered when you try to remove the tire from the rim and it splits wide open somewhere in the sidewall.
I had a Ford 1100 fwa compact tractor and the rears turf tires were so dry rotted that the rubber tread was coming off in chunks. I felt so bad about them when I agreed to sell it to a friend that I found him another set of tires in much better shape already on rims.
Your downtime needs to be considered, should the tire go, how long will it take to replace it, how much inconvience will that cause, lost money from a paying job, etc. If you use it strictly for your own and an unexpected delay isn't a big deal, then run them till they won't go any more. If your the type that requires a five layer hand wax job, over a lacquer paint job, then by all means change them now.
David from jax