Marion, I personally think they're jerking you around. I've never been made to check a fiddle. They fit in the overhead bins just fine, and fit under the seat if you can borrow part of the under-seat area of the one next to you. Most of a fiddle will fit under one seat. If they're telling you it won't fit, I'd ask what the dimensions of the overhead bins are.

I'd try almost anything before I let them stow it in the baggage hold - duct tape it to my back, take the fiddle out of the case and stick it in a shopping bag or very large purse, (it's my opinion that the fiddle will be safer without a case and in your hands than in the baggage hold) then let them stow the case, point out people carry gym bags, backpacks and various crates. Someone ought to design a fiddle case that looks like a large handbag or backpack for this sort of thing.

Now, very important: Loosen the strings!!! I also put something like a sock in between the strings and the violin body just in case.

If you don't, you'll end up with broken strings, a broken bridge, a snapped neck, or any combination. This is due to the pressure and humidity (possibly) changes. I had it happen once - the strings which hadn't snapped were far tighter than I'd tuned them.