Can road bumps excite frame eigen frequencies?
A resonant frequency encountered on a road surface would be for the whole bike system, i.e. frame, wheels, rider etc. The rider is effectively attached to the frame via elements with spring, damping and active control properties (arms and legs). Bumps encountered at a specific frequency might buck the rider off.

Bicycle frames are pretty stiff so any undamped resonant frequencies they have would be relatively high. Certain not at 10 or so Hertz you may get from riding on regularly spaced road bumps.

You assume that all structures fail when excited at resonant frequencies. This is plainly not true, otherwise acoustic musical instruments would self destruct when played. There certainly are structures where resonances can contribute to damaging oscillations of course, but I don't think bicycle frames suffer from this.