Yes, the voltage is the high end rating of the capacitor but the capacitor is for storing electrons measured in farads or microfarads.
If you forget about the technical jargon, think of it like a battery. Not quite the same but if you have a 24 volt battery supplying a circuit that has a cut off of 19 volts and you only charge it to 12 volts, you have a lot less electrons to supply your circuit than what is needed and chances are the circuit won't work.
A 25 \$\mu\$F capacitor that is rated at 16 volts will have a 25 \$\mu\$F charge when operated near the 16 volts but if you substitute a 25 \$\mu\$F capacitor rated at 35 volts you will not have 25 uf charge if you only apply 16 volts.
These capacitors have many functions in circuits. One main function is to supply electrons to a circuit when the normal plug in supply has dropped lower than needed such as with alternating current. As the voltage and current reverse, 60 times a second, the level goes from around 170 volts peak down to zero volts and on down to -170 volts and then it repeats. The capacitors filter this drop by supplying the appropriate voltage to keep the circuit smooth. As the voltage rises back up again, it recharges the capacitor.
A leaky capacitor has the effect of a large rated capacitor that leaks and keeps the circuit from working properly. In most cases, you can over rate a capacitor and get away with it. If you double the voltage value of the capacitor but keep the supply voltage low you might want to also double the Farad value. Ex: 25 \$\mu\$F at 16 volts to become 50 \$\mu\$F at 35 volts running on 16 volt supply.