My understanding is that it depends on what failed.
So like... if Test Track breaks down and your lightning lane reservation gets cancelled, you can redeem your selection for Test Track at a later time, or Soarin, or Living With the Land, or maybe Frozen, but maybe not Ratatouille. When I got one from a Figment breakdown, I was only able to redeem it at the crappy Epcot attractions that didn't need a lightning lane to begin with.
This restriction does not appear to exist at Magic Kingdom. There was one night where Space Ranger Spin kept breaking down, so we kept booking lightning lanes, which kept getting cancelled, and we used the experience redemptions to ride Thunder and Splash over and over again.