Get can be used to form the passive of verbs, and tire is a verb, so in theory got tired could be a passive of tire. But tired is hardly ever used that way : it is almost always an adjective, and so it is here. This is not "passive" get, it is "inchoative" or "transitional" get. "I got tired" means "I became tired".
With sick, sick is not a verb, it is an adjective, so this cannot be passive get. It is again "inchoative" get - I got sick is an informal way of saying I became sick. As far as I can tell, they are identical in meaning.
If instead of sick you name a particular disease, as in I got a cold (past tense of I'm getting a cold), this is another get - literally, to obtain, or acquire something. An alternative in this sense would be I caught a cold. (You can also say I caught cold without an article, but not I got cold in that sense, because the adjective cold would be undersood rather than the noun a cold, and we would be back to the inchoative meaning: I became cold, or I started to feel cold.