E.T, "to get sick" is an American expression. The past participle "gotten" is not used in British English. In your examples, #1 is correct. The tenses need to match. He could HAVE CAUGHT cold and he COULD HAVE GOTTEN sick. I get sick. I am getting sick...... present I got sick..... past simple I have gotten sick...... present perfect had gotten sick....... past perfect Of course, if you speak British English you will prefer to use the past participle "got", instead of gotten.

He could have caught a cold and got sick.