No. I will give an example. Let's say this person was ill for seven days. Were this the case, then 'I was ill' would refer to the whole week or to any time during the week. 'I became ill', on the other hand, would refer to the exact moment this person caught the illness.
I'm wondering if "Ich wurde krank" can mean both "I became sick" and "I was getting sick" - because "Wir wurden müde" is translated by Duolingo as "We were getting tired" .
In English, "I was getting sick" means something different than "I became sick." The first refers more to the development of the illness, while the second could simply be stating that I got sick (it had already happened).