How can the reverse voltage of an LED decrease?
LEDs shouldn't get damaged as long as they are used within spec.  If you didn't subject them to too much current in the forward direction or too much voltage in the reverse direction, then they should not get damaged.

However, many LEDs have a very low reverse voltage tolerance, like 5 V.  Depending on how your LED cube is wired, you may be subjecting some of the off LEDs to more than their allowed reverse voltage.  If these are blue or white, then even two LEDs forward may be more voltage than one LED can stand in reverse.

As always, check the datasheet.

For a more detailed answer, show a schematic of your circuit.