A question on Uniqueness solution of an IVP

Question

Let the IVP $(E)\ \ y'=f(t,y)$ with $ \ y(0)=1$ where $f$ is a function, which $f(t,y)$ and $\dfrac{\partial f}{\partial y}(t,y)$ are continuous on a rectangle $$R=\{(x,y): |t| \leq a, |y-1|\leq b\}$$ (so, there exist a local unique solution $y$ of IVP, which is well known by Peano's theorem). My question is: If we know that $ y_1(t)=3$, $t\in\mathbb{R}$ is a solution of previous d.e, can we prove with some statement that uniqueness give us $y(t)<3, \forall t\in \mathbb{R}$? The only thing I know for sure is that $y<3$, locally by continuity of $y$ on $0$. Maybe I don't express it very well but if you understand what I want to say, please any extra thoughts might be helpful. Thanks