Indeed:
Amtrak Bike Policy wrote:Folding Bikes Brought Aboard as Carry-On Baggage
Folding bicycles may be brought aboard certain passenger cars as carry-on baggage. Only true folding bicycles (bicycles specifically designed to fold up into a compact assembly) are acceptable. Generally, these bikes have frame latches allowing the frame to be collapsed, and small wheels. Regular bikes of any size, with or without wheels, are not considered folding bikes, and may not be stored as folding bikes aboard trains.
You must fold up your folding bicycle before boarding the train. You may store the bike only in luggage storage areas at the end of the car (or, in Superliners, on the lower level). You may not store bikes in overhead racks.
So you can bring them on all trains without restrictions (as long as their actually folding bikes). They just have to be stored in the luggage racks at the end of coach cars (cafe's/diner's don't have these, as I recall, so that's why they say "certain passenger cars."
No advance reservation or box or fee is necessary, just fold it up before you get on the train, stow it on the ends, and take your seat.
And come to think of it, I've taken my folding bike aboard
Northeast Regionals and Long Distance trains on several occasions without incident.