Physics A level: Photoelectric Effect

Hello,

I was wondering why electrons are not emitted when incident radiation is below threshold frequency. I thought that with enough photons, energy would build up in the electron allowing it to escape.

No, what you described is the 'wave theory' .
The energy of a photon is given by E=hf. Increasing frequency increases the energy of the photons.
Electrons have discrete energy levels.If the energy of the photons is below this energy,then the electrons cannot absorb the photons and therefore energy.They can only absorb the photons if the energy of the photons is greater or equal to the energy level of the electrons.
Increasing the rate of radiation(emitting more photons)won't do anything because the photons will get just bounced off if it has a frequency less than the threshold frequency.
Hope this helps!