Yes, that's right. You've understood what I was saying.
In the example She said that she would come but she might be late, you could interpret either that she said only one thing (that she would come) or that she said two things (that she would come and that she might be late). If you interpret that she said only that she would come, you could then interpret that the prediction she might be late comes not from her but from somebody else (perhaps the speaker.)
I assume the person who said it was wrong was making a point that if you mean that she said two things, the sentence would be clearer with the repeated that. In casual speech, however, the second that may well be omitted.
Here, I would like to say, for some conjunctions, such as but, so, or (to mean otherwise), and for, the second that should not be included at all. Here's my reason.
When we said:
I am not saying (that) he's stupid or that he's not working hard. The second that should be kept because we are talking about two separate things that are not related to each other at all.
Again, in this sentence:
He said (that) Jack got married yesterday, and that Mike was the priest. The second is again necessary because we are talking about two separate things that are not related to each other.
However, when using other conjunctions, separating two events might make the second part ambiguous.
For example,
He said (that) he put the cup on the table, so "that" he would not knock it over by accident. The second that makes the second part an independent clause, which is not meaningful without the connection with the first part. (You don't use "so he would not knock over by accident" alone without the first sentence led by that)
He said (that) we should do whatever she said or "that" she would kill us. Again, the "that" clause after "or" is not meaningful as an independent clause.
She said (that) she knew the answer but "that" she couldn't tell me right now. As you've mentioned, we could assume that the speaker says two things: That she knew the answer and That she couldn't tell me now"
However, if look more closely, we will find that separating what the person said into two independent "that clauses" makes the second part ambiguous again. The "but part" is only meaningful when considered to be a part of the whole "that clause" after "She said". (You don't use "but she couldn't tell me now" without the first sentence.)
Unlike Or (to mean alternative) or And, which simply shows an alternative or connects two independent sentences, "but (to mean however but as a conjunction)", "or (to mean otherwise)", and "so (to mean reason)", are conjunctions that entail connection with the first sentence and should not be considered to be an independent clause when used in the same sentence.