theres two type of suffering, the buddha said, mental and physical. yes a buddha can suffer physically, but will not suffer mentally.
look the arrow parabole:
"When touched with a feeling of pain, the uninstructed run-of-the-mill person sorrows, grieves, & laments, beats his breast, becomes distraught. So he feels two pains, physical & mental. Just as if they were to shoot a man with an arrow and, right afterward, were to shoot him with another one, so that he would feel the pains of two arrows; in the same way, when touched with a feeling of pain, the uninstructed run-of-the-mill person sorrows, grieves, & laments, beats his breast, becomes distraught. So he feels two pains, physical & mental..."
"Now, the well-instructed disciple of the noble ones, when touched with a feeling of pain, does not sorrow, grieve, or lament, does not beat his breast or become distraught. So he feels one pain: physical, but not mental. Just as if they were to shoot a man with an arrow and, right afterward, did not shoot him with another one, so that he would feel the pain of only one arrow. In the same way, when touched with a feeling of pain, the well-instructed disciple of the noble ones does not sorrow, grieve, or lament, does not beat his breast or become distraught. He feels one pain: physical, but not mental..."
This ^ is an absurdly overinflated, idealistic and impossibly unrealistic definition of "enlightenment" that puts it miles out of reach of anyone, and means that nobody ever becomes enlightened.
Beware of unenlightened definitions of enlightenment such as this one that make enlightenment impossible. Enlightened people can suffer just like unenlightened people can....
about the rest of what I said, its overall true. but since your right that its a over simplification that I was aware of, here the 7 factors of enlightenment
http://www.accesstoinsight.org/lib/authors/piyadassi/wheel001.html
Mindfulness (sati)
Keen investigation of the dhamma (dhammavicaya)[3]
Energy (viriya)
Rapture or happiness (piti)
Calm (passaddhi)
Concentration (samadhi)
Equanimity (upekkha)