Can the devil provoke bodily sicknesses?
Before anything else, it must remain very clear that sickness appear due to natural causes. To think that sicknesses have their origin in the world of spirits would be like returning to a prescientific state where reason would be substituted by myth. Now, if devils exist, we cannot exclude that sometimes they be able to act in this field. General rules, as their name indicates, are general, but nothing impedes that special happenings occur, however rare these may be. Normally only water or snow or hail falls from the sky, but on occasion a meteorite might fall as well.
In the same token, God can extraordinarily permit a devil to provoke a sickness. Saint Luke expressly mentions the case a "woman, who for the last 18 years had suffered from a sickness produced by a spirit and was bent doubled and quite unable to stand upright." (Lk. 13, 10-14). It isn't said if this woman was possessed, but it does say that the devil caused her sickness. This is a categorical statement in the Gospel. We could also add to it the case, in the book of Tobias, of Sara's different husbands, whose death was caused by the devil Asmodeus (Tob. 3).
St. Therese of Liseaux wrote a very interesting chapter in her life story:
The sickness that came over me proceeded, most certainly, from the devil. Since he was furious for your [St. Therese's sister] entrance into the Carmel, he wanted to take vengeance on me for all the harm that our family was going to cause him in the future, but he barely made me suffer; I was able to continue my studies and no one was worried about me. Towards the end of the year I started to have a constant headache. ( ) This lasted until Easter 1883. ( ) When I undressed, I felt invaded by a strange trembling. I don't know how to describe such a strange sickness. Today, I am sure that it was the work of the devil. ( ) I seemed to be in a constant delirium, speaking meaningless words (...) I was frequently faint, without being able to move the slightest. ( ) I think that the devil had received an exterior power over me, but he could not come near to my soul, nor my spirit, except to try to inspire great fear of certain things. (Story of a soul, chap. III)