The full picture is attained by solving for the modes of the whole waveguide (in theory for the refractive index profile right out to and beyond the jacket) by the methods described, for example, in Chapter 12 through 15 of:
A. W. Snyder and J. D. Love, "Optical Waveguide Theory", Chapman and Hall, 1983.
and the spectrum of effective indices ($c$ divided by the axial propagation speed of mode in question) for the bound modes lies between the maximum and minimum index of the refractive index profiles. So this translates to agree with George Smith's answer, for example, which gives a good intuitive description of the range in terms of a ray picture.
For many fibres, though, the range of bound mode effective indices is much narrower than the other answers would imply. This is because the modes in question are confined very tightly near the core, so only this region is relevant in setting the effective index and, especially for single mode fibres, the difference between core and cladding indices is miniscule. For a step index profile fibre, the fibre $V$ parameter:
$$V = \frac{2\,\pi\,\rho}{\lambda}\sqrt{n_{core}^2-n_{clad}^2}$$
where $\rho$ is the core radius, $\lambda$ the freespace light wavelength and $n_{core},\,n_{clad}$ the core and cladding refractive indices, must be less than the first zero of the first-kind, zeroth order Bessel function $J_0$, or about 2.405 if the fibre is to be single moded. For core radiuses that are readily manufactured (to wit, more than 1 micron), this means that $n_{core}$ and $n_{clad}$ are typically less than one percent difference. Let's plug in $\lambda = 1550\mathrm{nm}$ and $\rho=1\mu\mathrm{m}$ with $n_{clad} = 1.48$ (pure silica), then we find that the maximum $n_{core}$ we can have for $V\leq2.405$ is 1.59. This is an extreme example. More typically, for this wavelength, we would have $\rho = 5\mu\mathrm{m}$, when $n_{core}\leq1.4847$, a difference between $n_{core}$ and $n_{clad}$ of $0.3\%$.
So, for a single mode optical fibre, you can almost always say that the propagation speed is $c$ divided by the cladding index, and your error in assuming so will typically be less than half a percent.