After how many years does a carefully stored bamboo rod become unfit for casting/angling?

I guess it would depend on how well it was glued up and what kind of glue was used. I have at least two rods that were built around 1918 and I'm sure they were done with hide glue. I fish them at least a couple of times a year with no problem. If the rod has been maintained and there is no water damage, mildew, etc then it should be fine. I would worry more about the condition of the ferrules. Even well stored bamboo will expand and contract. this could cause the glue that holds the ferrules to separate or the ferrules themselves could crack(common with Montagues).
So the short answer I guess is Never.

That's a good question and would no doubt have at least something to do with the original quality of the rod. Provided it's been property stored and cared for, I'd think you might go better than 100 years. Maybe. A ton of actual use would reduce that no doubt, as the rod will soften with use.

While not the same thing, there are thousands upon thousands of wooden musical instruments still in use after well over 100 years that are still considered in fine playable condition. Granted, they don't do the same thing in the same way, but the material and adhesive technology of that time period hasn't given up the ghost, yet.

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I have bamboo rods made over 70 years ago that are still very fishable. Properly cared for they are a pleasure to use, from fly rods to boat rods. I refurbished an antique salmon bamboo rod for my father-in-law some years ago, and his grandson is now using it.

Heat seems to ruin them just as too much moisture does. I was told by my father many years ago that a bamboo rod should be taken out of its case and allowed to sit in a normal room temp and humidity for about a week before taking it out each season. I think he said that allowed the rod to absorb just a bit of moisture so it would be less brittle. Was he correct? I don't know but it sounded good to me!

One of the big things about bamboo rods is that you should never lean them against a wall or corner, because they will take a set, unlike fiberglass or graphite. If they do take a set it can be relieved with the judicious application of heat and bending back to the original straightness. It takes a lot of care to do so.