Advance train tickets

You can go to www.bahn.de and look up prices. You will see three priaces, two discounted ones, and one "flexpreis". The flexpreis is what you pay when you want a fully flexible ticket. It is also the price you pay at the station on the day of travel itself.

You do not need to buy tickets months in advance, but a few weeks in advance to get a discount is a usually a good idea.

Regarding Italy: I am not aware of any tickets getting cheaper last minute. Maybe you overlooked something.

You can go to the bahn website for you travel date and see the price of a nonrefundable ticket bought today (Super Sparpreis) and. Flex ticket, which is what you would pay at the station on your travel date.

I've been following advance purchase long distance train fares since early in the century (about 2008) when they started with seasonal (Sommer, Herbst, Winter, and Frühling) Spezial discounted fares. When the fares are announced, there are a certain number of the most discounted fares available. When these discounted fares (tiers) sold out, the next discounted (tiers) fares were higher (less discount). Unlike airfares, they didn't play games; it was straight forward. No, discounted fares never went down just before travel. That was for Germany; I don't know what the Italians are doing. (They don't even know)

When purchasing discounted fares on German Rail, buy your ticket as soon as you can be sure of making a specified connection. You can be assured that will be the lowest price.

Will the price of our train tickets really be much higher if we purchase them at the train station on the day of in Amsterdam rather then months in advance?

For German Rail tickets, usually, yes. Occasionally, all the discounted tickets are sold out long in advance, so there is no price advantage to ticket sale, say, a week before travel times. But usually, discount tickets sold even a few days before travel are still cheaper.

For trips to/through Germany I usually use a pass nowadays. These can be good value too, especially if you want to remain flexible.

I need to make several Interlaken - Amsterdam trips over the next months. A 1st class flexpreis return ticket is just shy of 1000,- euro!
I bought in stead a 4 days in a month Interrail, for 263,-....

Train ticket costs are a thing all of their own. But in this instance the route asked about appears that DB offers the least expensive price options if you buy one month in advance. I just picked one train. Others might be a little less or a little more but they all appeared to have their cheapest fares by purchasing one month in advance. Do your own study based on the train you want. See what you come up with. Also, time of year might impact how fast the cheap tickets sell as well. I guess, I would be looking about 6 weeks in advance to buy if it were me.

10:38 arrives 16:13 Duration: 5 hours 35 minutes
ICE123 and ICE1579:
Bought today (11th) From €166.60
ICE123 and RB68 (1579 is replaced for some reason):
Bought 18th From €109.99
Bought 25th From €99.99
Bought 1st From €76.99
Bought 8th From €37.99
Bought 15th From €37.99