It depends largely on the degree of the mutation. Most animals have a whole batch of DNA which is dormant, they are left in place from earlier stages of evolution and can, under the right conditions, become active again. This is why I hate it when people call stuff 'Junk DNA', it's not 'Junk', it just a mixture of dormant stuff and bits we don't understand.
When you get a big mutation, sometimes the organism is incapable of dealing with the degree of it and the system fails. A lot of 'Freaks' in Victorian times were mutations at extreme levels.
If you consider that every single one of us is a mutant to one degree or another, that it is, in fact, the engine that drives evolution, then I'm frankly surprised that these things don't turn up more. Or possibly they turn up more, but don't get reported.