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My Flight (with Adria Airways) leaves at 20:30 from terminal 1 of Frankfurt Airport. (On a Wednesday if it matters:)

I was thinking of arriving at the Airport from Göttingen with the ICE771 that arrives 19:16.

That gives me on 75 minutes before my flight to clear security and get to the correct gate.
Is this still enough time or should I go for an earlier train?

I will only have hand luggage and can check-in on-line before boarding the train.

Update: Considering that the ICE's are sometimes late, I bought tickets for an earlier train (one that arrives at 18:21). Better safe than sorry.

Assuming that Adria Airways closes boarding as usual 20-30 minutes prior to departure and you do check-in in advance, you have 45-55 minutes left for security clearance and finding the gate, what usually is more than enough time at Frankfurt Airport.
You'd also be fine if the train is 10 or 15 minutes delayed.

The important question that you haven’t answered is ‘where is your flight going?’ (intra-Schengen, extra-Schengen or to the US) Adria Airways suggests intra-Schengen, but I’ll focus on both points.

Depending on the time of year, whether it is one of the seasons in which German railways often experience problems (badmouting would say these seasons be winter, summer, spring and autumn), construction works and random other factors, the punctuality of long-distance trains is somewhere around 80 % — when DB defines punctuality as ‘less than 6 minutes late’. Going by the more helpful, more comparable and EU-wide definition of long-distance train punctuality (less than 15 minutes late), over 90 % of DB long-distance trains are on time. You can find the information on the official DB page.

Many airlines will suggest that you arrive at the airport two or even three hours before your flight leaves. However, especially for Schengen flights, planning to arrive at check-in one hour before departure can be okay. Take care to leave extra time if you are flying out of Schengen and especially when flying into the US! (although you can still get away with about an hour on lucky days flying into the UK, which is outside Schengen.)

Having an intra-Schengen connection time of 75 minutes is pushing it, but can work if you know your way around Frankfurt airport. Going out of Schengen I would definitely book an earlier train.

I personally frequently travel on the high-speed line southwards from Göttingen (although I usually don’t actually leave it towards Frankfurt) and daytime trains are typically only a minute or two late. (For some reason unknown to me, late Sunday evening trains manage to pick up much more delay, often coming in at around ten minutes late.) So personally, I would aim to arrive at Frankfurt airport station from Göttingen around about 90 to 120 minutes before an intra-Schengen flight.

Note that Frankfurt airport’s long-distance station is decidedly further out from the terminal than the regional station. If you happen to get a connection that makes you change onto local trains at Frankfurt central, that could well be a selling point rather than a disadvantage. (If by chance you arrive there early, you can take an earlier local train — S, RB, RE — too.)