eastwOOd86 is correct, that they were actually pretty strong. But they also had an advantage in actually not being that rigid. Their bolted-and-nailed wood frame construction ( [typically tough woods like ash and hickory, with non-splitting elm for the wheel hubs ) let them flex somewhat when being dragged over rocks and ruts. They were also dragged slowly- oxen were often used, and oxen do not gallop. And, as eastwOOd86 implies, wood is somewhat easy to reinforce, and repair, accepting nails and screws without too much of a loss in strength.

Well, for one, they were stronger build than you might think. The wheels are wood, but having wide wheels meant they could roll over larger ruts, limbs, rocks, etc. with less effort. The spokes are attached in such a way that makes a wagon wheel extremely sturdy. Added to that they have iron rims molded on the outside to protect the wheel surface. They also do have band springs in many of them to reduce bouncing on rough terrain.

But to more or less give another answer: they did fall apart. Virtually no one traveled a long distance in a covered wagon without carrying tools, king bolts, spare tongues, nails, even a spare wheel if there was space, axel grease, plus whatever else might be needed to make necessary repairs.

Long story short, they were very well-constructed, but still prone to break down and required repairs on long journeys.