Why would an input pin have both a pull-up and pull-down resistor?
I've done it several times when I couldn't figure out in advance which I really need.  So I would put both on the PCB and only solder down one.  In this way, if I was wrong I can just remove the resistor and solder the other one down.

These days, split termination isn't done so much but it used to be a popular form of signal termination.  To the unfamiliar, split termination looks like both a pullup and pulldown where the resistor values are usually less than 400 ohms.  

For analog inputs, sometimes an input needs a DC bias.  This can be done using a simple resistive voltage divider-- which also looks like a pullup+pulldown.  Normally in this case there is also something that blocks the DC, like a cap in series with the signal, before the resistors.

In my opinion, you never actually need a pullup and pulldown at the same time, since it just doesn't make electrical sense.  Using both at the same time will create a conflict and the end result is not an up or down, more like a pull-sideways!  :)  But there are lots of things, like the 3 that I mentioned, that will appear like a pullup and pull down at the same time.  These are very common, and I know of new engineers who confuse them for pullups+pulldowns.