The fact that vs The possibility that

The fact that is a common expression that we use when we talk about, quite obviously, facts, as in Copernicus first established the fact that the earth revolves round the sun. 

However, sometimes people erroneously use it when they make non-factual statements. Consider the following examples:

Was Woodrow Wilson oblivious of the fact that German nationalism might rise and create dire consequences...

How can I come to terms with the fact that I might never be in a romantic relationship?

The fact that if the price of milk, butter or cab rides increases, people might switch to soy milk, margarine, or the bus...

Now I am 23 and it is difficult to let go of the fact that maybe I'm wasting my potential as a good athlete.  

All these statements express possibilities or probabilities, as is apparent by the use of might and maybe; a fact is something about which there is no doubt. So, in these cases it would have been more accurate to say the possibility that or the probability that instead of the fact that.