My History professor once said, "If the past repeated itself, we should be able to predict the future." Keep this in mind as we delve into an example of what it means to be Ni.
Think about getting a test with the following math problem to solve 68 x 124 = ?. Unlike an Ni, I think a Si would suspend concluding on what the answer could be until 1) understanding the problem needs to be solved, 2) privately running through a calculation of some kind. Whereas an Ni could answer instantly by guessing on the spot before being asked. More often than not, an intelligent Ni could come to approximately the correct answer, because once the Ni was exposed to the test, he or she has actively taken in information, whether consciously or unconsciously, to anticipate the possibility of solving the problem.
The Si may read the math problem, and require a prompt "The test begins now" and more information or instructions to go ahead and solve the problem "Choose the best answer for each problem." However, the Ni continuously reasons over the smallest minutiae to form a basis for his or her knowledge. Even before the Ni receives the test with the problem on it, like when walking into the testing room, or the building where the testing room is located, information about the test would be rapidly poured over regardless of whether the information is confirmed. In this example the Ni receives the test and thinks 1) I have a test 2) tests need to be completed 3) completed correctly 4) there is a problem 5) with a question mark 6) questions need to be answered 7) calculating 70 x 120 can be done in my head 8) I can answer this as approximately 8400 9) the answer is probably 8432 10) my calculator confirms this answer 11) the instructions for this test tell me to choose the best answer for each question 12) the instructor told me "The test begins now"
The Ni takes in a barrage of information at all times, continuously processes this information to form conclusions about the tiniest of minutiae, which in turn become the basis for the Ni's knowledge about the world around him/her. So in essence, while the Si seeks external verification to make conclusions about information taken in, Ni relies on internal references to past memories, experiences, observations, and sensations to make (sometimes accurate, sometimes false) predictions about information taken in.