When solving a linear program via the simplex method, if the initial tableau is feasible then a different feasible point is generated after every pivot

Initially, I want to say yes; if a tableau is feasible then we will always have a feasible point after any pivot. However, the qualifier "different" makes me worried. Do pivots "generate" feasible points? I kind of figured the feasible points were just "out there" and putting the constraints in a tableau and using some method just determined the best feasible point.

Maybe I'm just misunderstanding the definition of feasible point? Anyway, any help much appreciated.