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.
I decided to do an experiment with a recently acquired KHS folder ("Hollywood"). 32" x 23" x 15". Bigger than the Brompton dual-hinged folder, but I could drag it like a carryon=wheels. I was flying AA domestic, out of LAX (757 jet). I couldn't get past the "guards" (working for AA) at the escalator, to get to x-ray machines. NO WAY.
I had to scramble to find boxes at the luggage store, buy tape, & box it up. It was oversize even for check-in (> 62" L+w+H). At check-in, I honestly answered it was a folding bike. They tried to invoke $150 charge!! It went back & forth, finally they cut me a break & only charged me $25 check-in fee (whew!!). I'm now flying back to LAX (AA regional carrier -> AA), with bike packed in smaller box (62" l+w+h).
Experiment failed. I ended up buying a cheap used dept store "mountain bike" for $10, for transportation at my destination. WIll be using this, as opposed to future attempts at using folder+travel.