Truncation error with growing step size

Yes, in $c_1h^p+c_2h^{p+1}+...$, the second term will dominate the first one for $h>\frac{c_1}{c_2}$.

In numerical applications, the many steps required by smaller step sizes eventually accumulate floating point noise sufficient to dominate the truncation error, so that a loglog plot of error vs. step size has a V shape with a fuzzy left leg, a middle piece on the right leg that is linearly raising and then some curved section for large $h$.

For a non-linear test problem $F[y]=F[p]$ with $F[y]=y''+1.16\sin(y)$ with exact solution $y(t)=p(t)=\cos(t)$ over the interval $[0,10]$ and using the 4th order classical Runge-Kutta method, this can look like this

The main error trends are first the accumulated floating point errors proportional to $\mu\frac{T}{h}$ where $\mu$ is the machine constant and $T$ the length of the integration interval, so that $T/h$ is the number of steps. And second the global error of the method proportional to $h^4$. A good fit was found with $h\mapsto\frac{10^{-15}}h+0.03\cdot h^4$.

Adding further higher order terms allows to reproduce the non-linear shape for larger $h$. Playing with the coefficients, a good fit was found manually with $\frac{10^{-15}}h+0.03\cdot h^4-0.08\cdot h^5+0.0225\cdot h^6$.