Sure, why not? Any process which imparts sufficient energy to the electron to kick it free of the nucleus is an ionization process. Heck, hitting the electron with another particle (while statistically unlikely) counts as ionization.
More than one photon can be absorbed, but the probability is minute for usual intensities. As a scale for "usual intensities" note that sunlight on earth has an intensity of about $1000\,\mathrm{W/m^2} = 10^{-1}\mathrm{W/cm^2}$.
The intuitive reason is, that the linear process (an electron absorbs one photon) is more or less "unlikely" (as the coupling between the em. field and electrons is rather weak), so a process where two photons interact is "unlikely"$^2$ and thus strongly suppressed. So for small intensities the linear process will dominate distinctly. The question is only, at what intensities the second order effects will become visible.
In the paper by Richard L. Smith, "Two-Photon Photoelectric Effect", Phys. Rev. 128, 2225 (1962) the photocurrent for radiation above half of the cutoff frequency but below the cutoff frequency is discussed. They note that for usual intensities the photocurrent will be minute, but that given strong enough fields
such as those observed in a focus spot of a laser (on the order of $10^7\,\mathrm{W/cm^2}$) the effect might be measurable. They also note, that thermal heating by the laser field may make the pure second order effect unobservable.
The more recent paper S. Varró, E. Elotzky, "The multiphoton photo-effect and
harmonic generation at
metal surfaces", J. Phys. D: Appl. Phys. 30, 3071 (1997) dicusses the case where high intensities (on the scale of $10^{10}\,\mathrm{W/cm^2}$) produce even higher order effects (and unexpectedly high, coherent non-linear effects, that is absorption of more than two photons by one electron). Their calculations explain the experimental observations of sharp features in the emission spectra of metal surfaces.
Historical fun fact: The 1962 paper is so old, that it talks about an "optical ruby maser"; lasers where so new back then, they did not even have their name yet.
Best Answer
As others have pointed out, the premise is false. However, there is still an element of truth to it, which is pretty easy to explain. It is true that when a photon has enough energy to ionize either a tightly bound electron or a weakly bound electron, it has a much higher probability of doing the former. This higher probability is shown by the fact that the K-shell edges are huge, constituing order-of-magnitude increases in the cross-section.
The reason for this is that you can estimate the cross-section using first-order perturbation theory, and it involves $|\langle i|eEz|f \rangle|^2$, where i is the electron's initial state (bound in an atom), and f is its final state (ionized). As discussed in more detail in the answers to this question, the transition matrix element tends to be small because the wavelength of a gamma or high-energy x-ray tends to be too small to be well matched with the spatial scale of the electron wavefunctions. You can say it either way around: the cross-section goes up with photon energy for a fixed electron orbital, or it goes down with electron energy for a fixed gamma energy.