Limiting voltage after potentiometer
If you want 250 mV to 4.75 V out of a pot using a 5 V supply, then put resistors on either end of the pot.  That's like having a pot that can't be adjusted to the ends.

Let's say you will use a 10 k&Omega; pot.  That 10 k&Omega; will span a range of 4.5 V.  The rest is simple voltage divider math.  If 10 k&Omega; drops 4.5 V, then 556 &Omega; will drop 250 mV.  Therefore, a 10 k&Omega; pot with 556 &Omega; resistors on each end will give you the adjustment range of 250 mV to 4.75 V that you want.