Is the following statement false because our solutions for the initial value problem may not exist when $t=0$; depending on our function? Also, uniqueness does not exist if the system is nonlinear, such as quadratic. Hence, the statement below is false.
Statement: Consider $x'=f(x)$ and suppose that for every initial condition $x(0)=x_0$ solutions exists and are unique for some time interval $[-a(x_0), a(x_0)]$ whose length depends on the initial value $x_0$. Then for all $y\in\mathbb{R^n}$ the solution of the initial value problem $x(0) = y$ exists for all time $t$.
Here is a counter-example: $$ f(x) = \sec^2(\arctan(x)) \quad \forall x \in \mathbb{R}$$ This function $f$ is smooth and so existence/uniqueness applies for the ODE $x'(t) = f(x(t))$.
However, for any $x_0 \in \mathbb{R}$ we observe that $$ x(t) = \tan(t + \arctan(x_0)) $$ is the unique solution to $x'(t) = f(x(t))$ for the initial condition $x(0)=x_0$. This solution is only defined on a portion of the timeline: It blows up to infinity in finite time.