A bhikkhu can refuse certain foods, for example those from the kinds of meat not allowable or for example, a bottle of liquor. The threefold rule does not require the monk to accept everything. Also, if food is offered at the wrong time, i.e., in the afternoon, it must also be refused.
Of courses, in some circumstances (such as those described by Ajahn Brahm in Isaan in the early 1970's) there is little choice, so eating frogs or whatever might be the only way to get enough food. However, that sort of situation is probably uncommon today.
I have often wondered about this one. AB mentions this event when the subject of eating or vegetarianism comes up. He reports that the whole frog was there in his bowl. Another monk stuck his fork into the frog and hit the bladder and urine spilled out. The other monk got disgusted and didn't eat. AB knew where the bladder was and was able to eat the frog in his bowl.
Since the whole frog was there in the bowl (not parts), couldn't or shouldn't he have "suspected" that the animal was killed specifically for him and the other monks? After all, it is the whole animal there in the bowl. I suppose it is possible the frog died of natural causes, but knowing that meat-eating and production is common in that area, it would seem that it is likely the frogs were killed specifically for lunch dana for the bhikkhus.
"I say that there are three instances in which meat should not be eaten: when it is seen, heard, or suspected that the living being has been slaughtered for the bhikkhu. I say that meat should not be eaten in these three instances. I say that there are three instances in which meat may be eaten: when it is not seen, not heard, and not suspected, that the living being has been slaughtered for the bhikkhu."
Majjhima Nikaya 55.5