Even if some bike were small enough to fit the airline carry-on size limit, there's also the issue that it leaves you open to the whims of the particular TSA agent. Their list of prohibited and allowed items doesn't mention bikes or bike frames, but it also says that it's not all inclusive. OTOH, it does prohibit any "tool" over 7" in length in carry-on and I could see an agent classifying metal parts of the bike frame as prohibited tools. Most probably wouldn't - but I don't think I'd feel comfortable taking the chance of running into problems.