It is usually said/done in textbooks and classes that if $\Delta x$ is known then $\Delta p_x$ can be estimated using the uncertainty principle as $\Delta p_x \sim \hbar/\Delta x$.
But the uncertainty principle does not say that, it says $\Delta p_x\ge\hbar/\Delta x$. That means we can only set a lower bound on $\Delta p_x$, i.e. $\Delta p_x$ could be anything between $\hbar/\Delta x$ and $\infty$
Why then the lower bound is chosen for estimation?
Are there certain situations where the products in uncertainties is of the same order as $\hbar$?