Jerry -
Your observation about the relative weakness of the threads in a wooden bottle stopper is 'bang on'. This can result in a failure when the blank is mounted on a threaded mandrel for turning. Also, when you cut threads with a tap, the tap rides on the threads that have already been cut as it advances into the hole. So depending on the timber you are using, it is even possible for the stress imposed by the tap as it advances to strip out the threads it has already cut. The basic cause for this problem is that stoppers are usually spindle turnings, so the hole is parallel to the grain in the timber, and threads cut into end grain are weak.
Some people use metal threaded inserts to avoid this problem (and some stopper kit suppliers include an insert as part of each kit they sell). You can also buy threaded inserts at a hardware store in either steel or brass (hint - steel is better - brass inserts are more expensive, and the soft brass can be damaged in the course of screwing them into a blank).
Another approach is to use wooden inserts with holes that are perpendicular to the direction of grain (ie, in 'face grain'). The result is that when the hole is threaded, the threads are much stronger. I normally make them from ash which produces very strong, clean threads in face grain, and as Bob suggests, I also saturate the threaded hole with thin CA to reinforce the threads. I make my inserts by drilling and tapping a series of holes in a scrap of face-grain ash, cut them apart, and then mount them individually on a stopper mandrel to turn them down to a final diameter (usually 5/8"). Then, when I make a stopper, I drill a 5/8 hole in the blank, and glue in a pre-made insert.