As matter approaches a black hole, does it speed up? If so, how do we know it speeds up?  Doesn't time slow down as gravity increases?  If time slows down around a black hole, is it possible matter doesn't actually speed up?
The answer is neither yes or no or possibly both.

Take a simple example. If something falls freely towards a black hole along a radial path, and is observed by someone who is far from the black hole, its velocity (according to the distant observer) is given by
$$v = -\left(1 - \frac{r_s}{r}\right)\left(\frac{r_s}{r}\right)^{1/2}c\, ,$$
(e.g. see chapter 6 of Exploring Black Holes by Taylor, Wheeler &amp; Bertschinger - freely available) where $r_s$ is the Schwarzschild radius and the negative sign just indicates an inward velocity with $r$ decreasing.

If you plot this function (see Fig.2 in Ch.6 of Taylor et al. - freely available) you will see that initially the magnitude of the velocity increases as $r$ decreases, but as $r\rightarrow r_s$ then $v \rightarrow 0$ and the falling object appears to come to a standstill (actually, because the light from the object is gravitationally redshifted, this may not actually be observed). However, if the velocity first increases and then slows to a standstill, then it must go through a maximum!

The maximum observed speed in this scenario is achieved at $r=3r_s$ and is $0.384c$.

Of course this story is different for different observers. If you are the falling object then your speed just keeps increasing through the event horizon and towards the singularity. On the other hand, an observer who was somehow able to hover just above the event horizon would measure the falling object's speed as just below $c$ as it passed.