We've traveled by domestic air carrier with our tandem in a hard wood case with no problem. We recently returned from a European tour with the same bike and box. Due to some fortuitous picks of airline and the complexity of airline baggage charges, it only cost us $75 extra for the roundtrip. Our bike was not damaged in any way.
Be that as it may, I started this thread as a warning to prospective European tourists. While waiting for a plane change in the Frankfurt airport, I watched the baggage handlers unload the bikes - there were several on this flight. This is how they did it: They positioned the baggage cart about 8' away from the end of the cargo belt. Then two handlers picked up each bike and threw them onto the cart, one atop the other, regardless of packaging. Not neatly, either, more of a pile.
During my tour I spoke to a German tourist whose bike was "destroyed" by Lufthansa baggage handlers. He has not been able to recover damages. His bike was in a plastic bag, as advised by a Lufthansa employee. When we arrived back in the US, we found that our bike had been traveling with the bikes of a couple who had put them in cardboard boxes. Their boxes were destroyed, torn and punched full of holes. She was not about to open the boxes at the airport, preferring to do her crying in private, so we don't know the fate of the bikes in the boxes.
Our box also collected a crack in the lid on the way back. They had thrown the bike so hard that it slid across the inside of the case and the captain's seat tube almost punched through the plywood box. One end of the rear hub was punched through the plywood, I can't imagine how. However, the bike was well-packed in the case and was not damaged. They also ripped the wheels off both our suitcases, one on the way over, one on the way back. I can't imagine how you can rip the wheels off a suitcase.
So if you go foreign, my strong advice is to have your bike in a rigid case, not cloth, not padded, not framed and padded, and not cardboard. Only a rigid case will survive this handling. And take cheap suitcases.
I was told that the baggage handlers are not employees of Lufthansa or any other airline, but rather low-bidder contract employees of the Frankfurt airport. I have no idea if this is also a problem at other European airports. I have not heard of this scale of a problem at any US airports.