While it may have sparked a few laughs when Uncle Bert fell over on that chair with the loose rung, this problem is usually irritating, and can be dangerous.
But maybe you’ve already tried to fix it, dabbing some glue around the rung and jamming it in the hole. Then, not long after, it came apart again. Have faith; it can be done.
The trick to repairing furniture is to remove all the old glue. To do this separate the broken parts as much as possible. You may need to tap the sections apart with a rubber mallet or a hammer wrapped with a some cushioning material. Don’t stress a joint by driving at an angle to the joint. Try and pull it directly out. If more rungs loosen, don’t worry, they’re ready to be rejoined anyway.
Clean off the old glue with a combination of chisels, scrapers and sandpaper. And get it out of the hole by carefully drilling using a bit which matches the hole’s diameter exactly.
Now, apply a bit of carpenters glue to both the rail and the hole. Insert the rail, and snugly tie a length of nylon line around the glued leg, and the other leg attached to the rail. (You may want to insert sponge pads between the line and the wood, to protect the finish.) Now create a makeshift “turnbuckle”. Take a short length of wood and insert between two lengths of line and twist it to tighten. Tie it in place.
By now, glue will have seeped out of the cracks in the joint. Wipe this off with a cloth dipped in warm water. Let it dry for 24 hours before removing the line. And invite Uncle Bert back over.