It can be installed on a lot of old dumbphones and allows to create a base station and mobile station. Might help you to understand the inner workings of the network protocol quirks.
Back then it was known to be _the_ project to easily build an IMSI catcher, so its scene got a lot of redteamers using it.
There is also an ongoing effort to reverse engineer the usb modem of the pinephone, but afaik it's still a lot of work. 
It's my understanding that the FCC and equivalents make it almost impossible for there to be open hardware that connects to the phone network. And any cracks they haven't filled, Qualcomm has NDA'd the documentation for.
When I worked at QCOM at one point I had access to that the 3g versions. My boss had printed it out. It was about 2 inches thick single spaced double sided. Rumor was the 4g one was double that. Then there was the 'chipset'/serial interface about the same size.
So could you re-write one? Yeah (given the time for it). Never mind the legal tangle that bcom, qcom, marvel, whatever Motorola is these days, Huawei, ZTE, and others have built around it. That would just be the technical. Then getting it onto a real cell network would take a lot of work. You prob could make your own in a unlicensed band though if you stuck inside the power limits. Title I/II determine what people can and can not do with the US networks. Think the current thinking is physical connection you can do quite a bit, but radio it is locked down. But I could be wrong.
Have to check that proj out. Looks cool!
LTE has some advantages over WiFi (i.e. the IoT features, the roaming features, the endpoint management features) so I can see a 2.4GHz LTE network being quite useful in some edge cases. With open technology, private LTE networks can probably serve quite nice business use cases.
It already ranges from 450MHz to 5.2GHz.
900MHz is also used for LTE in Europe, so I'm pretty sure you could do the same thing with 4G/LTE.
But yeah if you could keep inside the power levels, and stay in the unlicensed bands, and keep out of trouble with the patent courts you probably could make an LTE network. I would not be surprised if many of the chipsets already could do it. Power levels would be your biggest hurdle for something more than a test network.
There are lots of places that will make PCBs for you, and sell you the parts to solder together.
And run this on a Pi:
If you need something one layer deeper on the module or chipset level, there’s not really a lot out there.
In fact, I believe I have seen such devices for USB development - a configurable passthrough USB interface that would allow you to control what gets sent over the interface.
That kind of a setup would be much, much easier to develop and safeguard than developing a modem/baseband from the ground up ...
1. Much easier repairs since all schematics are known in advance.
2. Price. A lot of the price of the modems comes from licensing fees as others have already noted.
3. Integration. Directly embedding the pcb into your project will be much, much easier.
4. Collaboration. Working together in projects that utilize 4G becomes easier too. Many use cases might get revealed without the original creators intending for them.
5. Security. "More eyeballs on a line of code make bugs run away" and all that jazz.
Only disadvantage might be reverse engineer people might get bored not having enough things to RE :-)
Although a 4G module like a ublox is pretty easy to throw on a PCB, most of those things are just a couple chips inside so it just seems like there's not a lot of repairs other than just swap out the module. I've seem those modules in the $20 range, seems like a lot of work to get around the licensing fees on a $20 module?
Some creative googling got me this, for a different starting point. Looks like it might be a useful project writeup dealing with SDR and connecting to some cellular sources.
I'm not sure how common open SDR hardware is, though some forum threads seem to suggest the Hackrf schematics are here, and are meant to be replicated:
I think one of the big problems is ultimately going to be dealing with phone companies. I think that will ultimately stymie any open efforts in this space, since at the end of the day, you need someone to accept your cellular traffic.