As the other answers already noted, there are cases where the Newton iteration does not converge.
One interesting question is for how "many" values this can occur and whether it's a set or full measure or a Null set. As it turns out, there are indeed polynomials for which the Newton process does not converge, and the set of such starting values is a set of full measure.
This won't occur for polynomials of degree less than 3, but it may occur for cubic polynomials like
$$
p(z) = z^3-2z+2
$$
The dynamics of the Newton process $z\mapsto {\cal N}_p(z)$ is made visible by the graphic below which displays the dynamic for real starting values with $z\in[-4,4]$.
The image consists of 18 colored stripes, each color encoding a real number (−4=purple, −2=violet, −1=cyan, 0=black, 1=red, 2=orange, 4=yellow, ±∞=white). From top to bottom, the stripes show values of the $n$-th Newton iterate, where row 0 indicates the starting values.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/dKBB1.png)
Starting values smaller than approximately −0.83 converge to the violet zero of $p(z)$ at $z\approx-1.76929$, but all values in some interval around $0$ are attracted by the black-red cycle
$$\cdots0\mapsto 1\mapsto 0\mapsto 1\mapsto 0\mapsto1\cdots$$
But how many such "bad" polynomials are there? Are such polynomials a null set in the set of all polynomials, or is there a substantial amount of such polynomials?
For cubic polynomials one can study the behavior of the critical point of ${\cal N}_p$. For example, $0$ is a critical point of ${\cal N}_{p_\lambda}$ for all cubics from the family
$$
p_\lambda(z) = z^3+(\lambda-1)z-\lambda
$$
Plotting how the critical point of ${\cal N}_{p_\lambda}$ behaves under iteration gives the following result, where $\lambda$ ranges over the complex plane where $-2.5\leqslant Re(\lambda),Im(\lambda)\leqslant 2.5$.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/zdbO2.png)
There are tiny black spots that are subsets of full measure in the $\lambda$ space for which the critical point does not converge to a zero of $p_\lambda$.
Magnifying the black spot around $\lambda\approx0.3+1.64i$ looks like this:
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/8xEYo.png)
There are infinitely many of them.
Answer to a comment: The hue in the last two images represents the limit of
$$\lim_{n\to\infty} {\cal N}_{p_\lambda}^n(0)$$
i.e. the fate of the critical point 0 of ${\cal N}_{p_\lambda}$ under iteration, provided the iteration converges.
When it does not converge, then the point at $\lambda$ is colored in black. There are 3 hues around the island in the 3rd image because $p_\lambda$ has 3 complex zeros that may occur as limit. The hue indicates which zero and relates to the limit in a non-obvious way and such that the 3 hues are 120° apart in color space (red, green, blue). This comes with a grain of salt because the exact locations of the zeros of $p_\lambda$ depend on $\lambda$, so this only holds approximately and only is small regions. It's the reason for why the colors of the zeros in the overview picture transition smoothly over large stretches of the $\lambda$ plane.
The brightness indicates how many iterations where required to conclude that the iteration converges for that $\lambda$: brighter = more. For points close to the Mandelbrot islands, the saturation is increased again before it finally drops to white. This is for aesthetical reasons and results in the purple-white glow around the black islands.
The algorithm smoothes out the colors by interpolating the number of iterations (which are natural numbers) to real numbers; similar to how you would smooth the potential around an electrically charged Mandelbrot set. To that end, one has to introduce a notion of distance to the closest complex root of $p_\lambda$.
It's more than 10 years ago since I make these images; so bear with me. The summary of the wiki page contains and explains the relevant part of the C code I used back then. The idea for this specific family of cubics is from the book of Peitgen and Richter.