I know that, in most of the world, phone conversations are transmitted as digital data (whether frames in circuit switched protocols or packets). I know that several companies offer products which will convert GSM conversation data to VOIP at the base station or MSC level. I assume this data is not archived or saved anywhere (unless requested from authorities to the phone company). I think (but am not sure if this reflects today's world) this data is not encrypted for most of the travel from A to B (in GSM it's actually encrypted from the handset to the base station). I do know that most of the network infrastructure is reachable via IP from inside the owner company for management purpose.
My question is: is it technically possible for bad guys (or governments) to infiltrate from outside the routing infrastructure (without the phone company's knowledge) and archive / copy not just metadata, but conversations of one or more specific persons? Has this happened in the real world so far?
Giankun, this story by The Intercept says that it has already happened, as Steffen Ullrich said in the comments to your question. The link i provided above tells the long story about it. This second link talks about the greek incident and about another one that occurred in Italy.
There lots of articles about these incidents. Search for the names of dead engineers and results will pop up. There is also this blog entry titled "Who's murdering NSA whistleblowers?".
As you will note, officially, the mentioned institutions have no relation with the reported deaths, but the articles show that your question is already a reality.
Yes. One of the more famous examples of this happening is phone phreak Kevin Mitnick. 20 years ago Mitnick was able to break into the telephone network, and tap the phone conversations of the very agents trying to catch him. Security is (hopefully) a bit better now, but there's no real reason to believe it's not possible to do this now.
The short answer - yes and yes. It's been done before by governments by breaking into ISP infrastructure, wiretapping at traffic aggregation points etc (just say smth about terrorists or child porn and you're free to break any law you want).
The longer answer about GSM: all the means of "over the air" interception I now of can be (in theory) detected by mobile operator if they carefully monitor their network for anomalies (flood of RACH disabling the cell, heavy radio interference, appearance of unknown cell in mobile reports etc). In practice - I doubt they bother.