When we brought our Brompton to the boarding gate for our first flight, without concealing that it was a bike, the agent was upset that we did not bring the bike to the check-in gate and did not pay the bike fee. In the end, he let us onto the flight though without further action. That is the only piece of own experience I have on that.

Needless to say that the bike fee (of $150 according to the Delta site rather than $200 which the agent claimed) would have been a major hit. The Delta site further talks about the bike as a checked item. The bike's rectangular dimensions of 23x23x12" (I hope I don't make a mistake in the latter) exceed the nominal Delta limits on a carry-on. I am saying nominal, because 23x14x9 appears violated by a big portion of carry-ons that get on board. I am not sure one could put down a blanket statement on what an agent were likely to do. In our two further experiences, the agents were satisfied when the items appeared to belong to a class accepted as carry-on without constraints other than getting through security.

As to flying from Europe, I've been roughened enough many times even when having all the regulations on my side, so no I am not likely to seek extra opportunities.