Are gravitational time dilation and the time dilation in special relativity independent?

Let me try and expand a bit on Ben's answer.

Starting with special relativity, the key thing to understand is that all the weird stuff, and indeed the Lorentz transformations, is derived from a property called the metric. If you have two points in spacetime separated by ($\mathrm dt,~\mathrm dx,~\mathrm dy,~\mathrm dz$) then the metric tells us how to calculate the interval between them. For SR this is:

$$ \mathrm ds^2 = -\mathrm dt^2 + \mathrm dx^2 +\mathrm dy^2 +\mathrm dz^2 $$

The interval $\mathrm ds$ is referred to as the line element and is an invariant, i.e., every observer no matter how fast they are moving, will calculate the same value for $\mathrm ds$.

The equation for the line element should remind you of Pythagoras' theorem, and indeed the only difference is that the sign of $\mathrm dt^2$ is negative not positive. It's this difference in the sign that is responsible for effects like time dilation. This is the important point to take home: this metric is all you need to calculate time dilation.

Now consider general relativity, and the effect of gravity. But first let me rewrite the special relativity equation for the line element in polar co-ordinates:

$$\mathrm ds^2 = -\mathrm dt^2 +\mathrm dr^2 + r^2 (\mathrm d\theta^2 + \sin^2\theta~\mathrm d\phi^2) $$

and now I'll write the equation for the line element near a black hole, i.e. the Schwarzschild metric:

$$ \mathrm ds^2 = -\left(1-\frac{2M}{r}\right)\mathrm dt^2 + \frac{\mathrm dr^2}{\left(1-\frac{2M}{r}\right)} + r^2 (\mathrm d\theta^2 + \sin^2\theta~\mathrm d\phi^2) $$

If you compare these two equations it should be immediately obvious that they are very similar, and indeed if you let the mass of the black hole, $M$, go to zero or if you go a long way away, so $r \rightarrow \infty$, then the two equations are the same.

This means the GR metric includes everything that the SR metric predicts, but it adds to it. So there isn't a distinction between the time dilation due to just velocity and the time dilation due to gravity. The GR metric is an extension of the SR metric and includes both. However let me reinforce Ben's cautions: it generally isn't useful to try and separate the time dilation due to velocity and the time dilation due to gravity.

It's not true that gravitational time dilation is based on the strength of the gravitational field. By the equivalence principle, the gravitational field equals zero for any inertial observer, and can have any other value you like for some other appropriately chosen observer. Gravitational time dilation is based on the gravitational potential.

Neither kinematic nor gravitational time dilation requires general relativity. You can have a gravitational field and a gravitational potential in flat spacetime, e.g., for an observer inside an accelerating elevator. There are straightforward special-relativistic arguments that derive the gravitational time dilation from thought experiments involving accelerating elevators.

Neither kinematic nor gravitational time dilation is fundamental. What's fundamental is the metric. Either effect can be calculated from the metric.

Gravitational time dilation can't be fundamental because the gravitational potential isn't even well defined unless the spacetime is static. For example, cosmological spacetimes aren't static.

A good popular-level book that explains the fundamental status of the metric very clearly is Geroch, General Relativity from A to B.