Train transfer time only 4min. in Nürnberg Hbf. Is it conceivable?

Here are some maps of Nürnberg Hbf: http://www.bahnhof.de/bahnhof-de/Nuernberg_Hbf.html
The maps don't show the numbers of platforms, but it looks like they are in a simple row connected by a tunnel.
However with only 4 min, you don't have any leeway in case your train is only slightly delayed.

I found a station plan here: http://www.stationsdatenbank.bayern-takt.de/StationsdatenbankBEG/picture/image.html;jsessionid=99AA8C0C7100DDCB058858CA004B6512?imageId=1276615
It is a just a matter of searching in German. It looks like all platforms (= Gleis on the plan) are parallel, so it is just matter of going down, across and up.
Even so, 4 minutes sounds very tight to me, especially if you have luggage. 10 minutes would be more usual.
I would check when the next train to Rothenburg ob der Tauber is, probably 30 or 60 minutes later. Then, if you miss the train you just go and have a coffee or a beer.

That is very tight, even though Nuremberg station isn't as large as some. But if you miss your connection due to delays, DB will still honour your ticket.

The tracks at Nürnberg are all arranged in parallel, with a tunnel underneath connecting them. It's simply a matter of getting off the train, going down to the tunnel, and going back up to your desired platform. Once you see the layout, you'll realize how easy it is.

Even if for some reason you miss your connection, this shouldn't be a problem. Unless I'm mistaken, the next leg of your trip to Rothenburg odT is on a regional commuter train. Which means that your ticket is valid on any train on that route, not one at any specific time. If you miss the first, at most, you'll probably only have to wait 30-60 minutes for the next.

I'm looking at the schedule for May 3, and all of the connections between Gleis 6 and 18 on the way to Rothenburg is a 6 minute connection, from an ICE to an RE train. That should be doable, just be standing at the exit door of the train as it pulls into Nuremberg, push the door-open button the instant the train stops and move to the stairway down to the tunnel. You should make it.

Exactly. Make your way with your luggage to the exit door before the train stops and be the first people to get off the train.

With that said, why not just get an earlier train to Nurnberg and have a bit more time to make the connection?

While not ideal, I've made similar connections before in Germany. Like others said check the station map and how you incoming train will be oriented in the station and also be prepared with all you luggage before you arrive you you can jump out the door and strait to you next train. And even if it turns into a comedy of errors and you miss it you won't have to wait too long for the next train and it might make a good story!

DJ

Nuremberg is an excellent place to miss a connection. You should plan to do so.

Old town near the castle
Nuremberg has some great pubs
Even if you have less than an hour there, you can visit the Handwerkerhof near the station.
Nuremberg sightseeing

What Tom said is right - if you miss your regional train connection at Nuremberg, no worries - you can take any other regional trains you like that same day to Nuremberg. (If you miss a high-speed train connection you're out of luck unless it was DB's fault... then you have to get your ticket authorized to continue on the next high speed train.)

Don't be intimidated by the span between platforms (6/7 to 18/19). Most of your time will be spent getting off the train, to the stairs, down the stairs, up the next stairs, and to the train. The actual time in the tunnel will not very much longer than for a one platform change. Nevertheless, this is a tight change, particularly if the ICE from Munich is even a few minutes late. The main thing is - there will probably be others making the same change - stay with the herd. The conductor will not release the train when there are still people boarding.

I'm assuming you have non-refundable, advance purchase, Savings-Fare tickets. According to the vor- und nach-Lauf provision of the Conditions of Carriage (AGB) of the Bahn, you have until 10 AM the following morning to complete your trip from Nürnberg to Rothenburg odT on any regional trains on the same route (ie, probably, Nürnberg to Ansbach to Steinach to Rothenburg). And, if you have Standard (Full Fare) tickets, they are still valid at least until the end of the day. So don't panic if you miss the connection in Nürnberg. Just take the next regional train to Ansbach; there's a regional express (RE) leaving every two hours from track 18, and S-Bahns every 20 minutes from track 22. There's a regional train from Ansbach to Steinach every hour.

This doesn't help you now, but might help others. I would not have purchased tickets using the ICE to Nürnberg. This route saves you up to an hour (if you make the connections), but depending on the age of your children, cost you 48 or more euro (each way, for Savings-Fare). You could also go by regional trains (Munich-Treuchtlingen-Steinach-Rothenburg) with only two train changes in very small stations, and you could use a Bayern-Ticket, purchasable up to travel time, 28€ if both children are under 15, 38€ if not.

Even if you have less than an hour there, you can visit the Handwerkerhof near the station.

Or at least buy some Lebkuchen from a street vendor to enjoy on the next leg of your trip! :)

I had a 5 minute connection at Kreiensen Bf, from platform 1 to platform 52. Despite how it sounds, it was very simple - I made it with 4 1/2 minutes to spare.