If you look here, on slide 24 the quadratic variation of a scaled random walk comes out to be t (the n doesn't matter as it cancels out).
A scaled random walk becomes a brownian motion as n approaches infinity.
So, why do we say that its t with probability 1 when we can make a stronger statement?
Alternatively, if that's not the case, how can we construct a path that can be a realisation of a brownian motion but where the quadratic variation isn't t?