Probability that the sum of 'n' positive numbers less than 2 is less than 2

This of course depends on how you choose the numbers.

We could assume $X_1$, $X_2$, $X_3$ are independent random variables each uniformly distributed on the interval $[0,2]$.

I'd rather consider $Y_1=X_1/2$ etc., and ask for the probability that $Y_1+Y_2+Y_3<1$. Then $(Y_1,Y_2,Y_3)$ is a point uniformly distributed in the unit cube $C$ (with vertices $(0,0,0)$, $(1,0,0)$, $(0,1,0)$ etc.) Of course $C$ has volume $1$, and the probability you seek is the volume of the tetrahedron $T$ with vertices $(0,0,0)$, $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$.

Using the formula that the volume of a tetrahedron is a third of the area of the base times the height, then this is $1/6$.

In $n$ dimensions the probability is the $n$-volume of an $n$-simplex with vertices $(0,0,\ldots,0)$ and the $n$ standard unit vectors. You can compute its volume when you know that the volume of an $n$-simplex is $1/n$ times the $(n-1)$-volume of a face times the corresponding height.