I would say it would have to depend on the size of the instrument in question. I cannot imagine that a Tuba or a Cello would be "carry on".For me, I have yet to take my instruments on a plane with me. When I have flown to gigs, there has always been at least one from the band who has driven. I send them along with them.
I would say that, a *good* case...like the ones they pack cameras in with all the foam rubber...would be my choice if I had to take something precious with me, especially if I had to check it.
I've seen what they have done with my regular luggage! :::SHIVER:::