The voltage rating on a capacitor is the voltage level above which it is likely to explode and get very messy.
The voltage doesn't affect the capacitance in any way.
As long as you stay well below the rated current all will be fine. The closer you get to that rated current, the more likely you are to run into manufacturing tolerances and defects and have a capacitor explode, or vent. I for one always use at least double the required voltage if possible.
Many modern motherboards have capacitors that are rated at just over the required voltage, and it is amazing how many motherboards die with vented caps. You can always tell a vented cap - the top of it is bulging outwards.
For more information: Capacitor plague - Wikipedia