Can "have a feeling that" be used to state a fact?

The sentence "I have a feeling that George Washington was Commander in Chief" is not equivalent to "I know that George Washington was Commander in Chief" nor "George Washington was Commander in Chief".

In the first sentence, you are stating that you have a belief about a certain fact. The fact may be false. Sometimes this form is used to correct information:

A: "So George III was the Commander in Chief of the Continental Army."

B: "Well, I have a feeling George Washington was the Commander in Chief."

In this example B is correcting A using a less-strong assertion of the facts, however the implied meaning is "George Washington was the Commander in Chief".

The sentence "I know George Washington was Commander in Chief" asserts the speaker's knowledge of the fact. The sentence "George Washington was Commander in Chief" merely states the fact.

Sometimes the "feeling" form is used when the speaker is unsure about the facts

"I have a feeling it's going to rain today."

Compare this to

"It's going to rain today."

The second form asserts it as true that it will indeed rain today. The first merely states the speaker's belief, and suggests that the speaker acknowledges some doubt.

Sometimes people say this about something they know darn well although it's not an objective fact.

"When I saw the jurors faces, I had a feeling the defendant was not going to like their verdict."

Using this phrase to correct someone strikes me as obnoxiously sarcastic: "Well, personally, I had the feeling this was Tuesday, but you're entitled to your opinion." Unless the fact in question is somewhat obscure, eg, the capital of Assyria, or the average airspeed velocity of an unladen swallow.