This claim appears again and again, and [I hope I may be forgiven for pointing this out] it is false.

If you had a mosquito [even a heavy one] jumping up and down on a sturdy, thick aluminum frame [a Santa Cruz Bullet, for example], the frame would not reach its fatigue limit in your lifetime, my lifetime, or all of our lifetimes put together, or within any other reasonable time frame -- and not within most wildly unreasonable time frames.

[Perhaps there is a metallurgist out there who could tell us when and how failure might eventually occur -- I suspect that these materials slowly degrade or decay over the course of hundreds of millions of years. Assuming an immortal mosquito, I doubt if its many millennia of activities would be the primary cause of the failure.]