Below are a couple of hypothetical graphs of the response of a bike to being acted on by a driving force at a particular frequency.
The unrealistic one on the left is the kind of graph that would lead to the pop-culture scenario where the bike breaks apart due to resonance. The resonance has to be at a pretty low frequency to match the frequency of the bumps in the road. There is very little damping (internal friction in the bike-rider system), and this shows up in two ways: (1) as you'd expect intuitively, the response is higher because energy isn't being dissipated rapidly into heat; (2) less intuitively, it makes the resonance peak very narrow.
The second graph is more like what I would expect for a system like a bike frame coupled to a rider's body. Because the frame is a complicated object, it has many resonant frequencies, not just one, and each one is fairly weak. I expect that the resonant frequencies would be rather high, probably 100-1000 Hz, and higher, because the frame is stiff and light. There is a lot of damping, due to coupling of the frame to the rider's soft squishy body, coupling to the tires, and possibly also due to internal friction in the frame. This damping makes the peaks not very tall, and also makes them wide, so they blend together.
If you want some experimental evidence about this, take your bike and bang on it with your fist in various spots and in various directions. (I guess you want to do this while sitting on it, since your body affects the system with its mass and damping.) If there are resonances in the audio region, you'll hear those frequencies as audible sound. If there are low-frequency resonances, you'll feel those as vibration. You can tell the amount of damping based on how long the ringing sustains. If it rings like a bell, you have low damping. If it's a quick thud or "dink" that immediately drops off, you have high damping.
Even if you assume the worst-case scenario of the left-hand graph, the peak is very narrow. (To make it tall enough to destroy the bike, we have to make damping low enough so that the width becomes very small.) This means that the hazard would occur only at a very, very specific, small range of frequencies. I think if most of us experienced this while riding, we would instinctively slow down a little. That would cause the driving frequency to drop and immediately take us too far from the resonant frequency to excite any significant response.
Designers clearly would not want to make a bike that would vibrate with low damping in response to a vertical force from a bump in the road. Such a bike would oscillate up and down for a long time, which would be annoying. Even in the case of a mountain bike that has shock absorbers, there is probably a ton of damping designed into the shocks in order to avoid just this kind of effect.