I recently saw a sentence like this:
I thought that I can't do it alone, but he convinced me that I should at least try.
I am almost sure that we can't remove the second 'I' and change the sentence to:
I thought that can't do it alone...
but someone argued that it's possible. I thought about it a lot but couldn't come up with a grammatical explanation to prove that we can't remove the second 'I', despite the fact that I don't remember seeing such a sentence and it just seems wrong to do it.