I had an Euro rail pass and the only train late was the one connecting me with my expensive night train accomodatins to Venice. It didnt wait and I made it with 30 seconds to spare
To add to John's advice, allow more than a few minutes to make a connection. It can take 5-10 minutes just to get off the train because of people ahead of you with luggage. I missed a connection in Milan on my Paris-Venice trip. The first train was a few minutes late and it took a few minutes getting off the train. Then I ran thru the station trying to get to my next train. It had already left the station. I think the schedule had allowed about 20 minutes for this transfer, but the late first train left me looking for an alternate train to Venice. I ended up catching a train to Verona and then to Venice. This glitch added two hours to my trip. Be prepared to make adjustments in plans. It helps to have an idea about possible alternate routes, and not be loaded down with lots of luggage.
Definitely a good reminder to check out the other possible connections, in case of a late train. I usually print out some earlier and some later connections -- and bring them along.