Well, I used to work for Delta as a mechanic and have flown quite a bit with a bow (I was able to fly for free). But more importantly, I've seen thousands of planes loaded and unloaded and I've seen ramp operators in action where passengers don't. 99.9% of the time all goes well but when things get hectic, accidents happen. It was a running joke at the airlines that any letter written by an employee to explain an accident or incident that put him question always starts with "While rushing to avoid a delay...." It's a very fast paced job at times so, $h!t happens.
A couple things you can do to help ensure the safety of your bow are:
1. Always use the most heavy duty case you can afford/find. I've had two of the heavy schedule 40 pvc cases busted. Both times the bow was ok and the airlines paid for the price of the case. PVC and the fittings/endcaps are not cheap and it takes close to $40 to build one
2. Always put your bow in a bow sock and then find a big piece of soft foam like an old sofa cushion. Cut two circles about twice the diameter of the tube and shove one of them all the way to the bottom as a cushion on that end. Then, cut a slit in the center of the second piece and slide it onto the top limb of your bow about 15 to 20 inches. Slide the bow into the case and when you get to the foam on the top limb, compress it around the bow and force it into the tube. That will keep the bow centered in the tube so it doesn't flop around. It also serves the more important purpose of keeping the bow snuggly in the tube if an end cap gets broken. Both times my tubes were broken it was on the ends but since my bow did not come out of the tube, it survived. Put a third piece of foam over the top end before screwing the end cap in place.
3. When checking in, request to have it "hand carried" or "gate checked". This is common for skis and fishing poles. There may be a small charge but not likely most of the time. If it is gate checked, when you come out of the plane into the jetway, to should be waiting right there for you in the jetway. Gate checked items are the last on the plane and the first off. The ramp operators usually have them out of the cargo and into the jetway before coach starts to deplane.
4. Modify your tube so that it can't roll. You can use two blocks of wood cut to triangle shapes with a hole bored in them to fit over the tube and glue them in place before the end caps are put on (If making your own) or you can cut 3 pieces of wood like an old broomstick and use glue or heavy tie wraps and fasten them to the outside. Evenly spaced, they form 3 "corners" and the tube will not roll. If you have access to a strap banding tool, that works great. This is important because a tube that rolls can roll off the baggage belt loaders and get run over or get stuck in the airport terminal conveyor system, cause jams, get broke etc. I once watched from the terminal window while waiting to board and saw the rampie driving a belt loader to the plane with my bow tube on it and as he turned, the tube rolled off and right in front of a tug pulling a bag cart. Luckily the other guy stopped in time.
And, as someone else pointed out, sending it via mail carrier avoids pretty much none of these issues. It all has to get shipped by plane.
Also note, I've been out of the industrie for about 3 1/2 years and security changes may have occured that prevent gate checking at some airlines.