Graham, where will you be travelling? And, will you be continuing with your box from place to place, or leaving it in one spot & returning to it?
VIA Rail Canada, for example, is very good with providing big cardboard boxes free of charge. The downside is, er, that the rail network itself is less than comprehensive in its coverage... Can't comment on Amtrak in the States.
My recent experience in Europe last Sept/Oct fits with Pete's: a large cardboard shipping box from a bike shop works well all round, both for air travel and for land travel. My experience with European train
stations, however, was far from easy: no porters & no trolleys! So, if you're transporting bike-in-box plus panniers, etc., you've got an awkward handful.
You may want to check out my journal on CGOAB on this, plus some useful comments on the Guestbook from another rider with a bike-in-a-cardboard-box in Europe, and the storage of these things. Here's the link, with the stuff about bikes in boxes & on trains towards the end:
http://www.crazyguyonabike.com/doc/2012Rhine-DanubetripSome friends use the expensive wheeled hard boxes: these are protective, and easily managed, but cost more than some bikes. Then, there's the storage issue: when you reach your point of departure for your ride, what happens to the hard case?
Hope this is helpful -- good luck!
J.