I did it with the family, AMS-FCO (Rome) in 2011 using ID90 paper tickets. (it probably has changed, lots of airlines went to eTickets) Went smoothly, I was able to check at in AMS via a kiosk (I called KLM and made a listing prior to leaving the states), got their equivalent of Seat Request cards. Went through security no problem, checked in at the gate. They called our names probably 10 minutes prior to departure - I asked them at about 25 minutes and got a haughty, "we'll call you".

This was in February, so it wasn't really busy so I can't tell you what summer would be like, but I get the feeling they don't overbook like US airlines do. European airlines have high levels of compensation they have to pay for denied boarding.