Firstly, you ask whether it is possible  that spectrum is infinitely divisible. Let us dispose of that first. Not too long ago we discussed concepts relating to physical infinities. For excellent reasons everyone agreed that there is no such thing as a physical infinity. There are two aspects to this in this connection. For simple reasons of quantum mechanics, the spectrum is not infinitely divisible. The more closely we examine light, even laser light, the more the measure of its wave length blurs. The other aspect is that even if it were infinitely divisible, we could not infinitely divide it. To do so we would need infinite precision of measurement, and we are nowhere near that. Last I heard, only specialist physical laboratories could do much better than 10 decimal places. 10 million decimal places would still not be infinite procession, and I am absolutely confident that no human measurement of light wavelengths will ever get near even 100 decimal places.

You also ask whether  technically bandwidth could be more finely utilized than is currently the case. No doubt it could and will be, but it is called bandwidth precisely because you need a certain channel width if it is to bear your message, and a certain separation from the next channel if you are not to suffer crosstalk problems.

You mention the likes of compression technology or multiplexing. They have been developed incredibly in the last century or so, and are very well understood. The available bandwidth has increased by many orders of magnitude in my own lifetime, using increasingly advanced control of frequency, multiplexing, noise control and lately of digitised logic. The sheer volume of data transmission now available would have seemed nonsensical even fifty years ago. Even satellite microwave transmission held promise of more bandwidth than anything we could envisage needing. The capacity that fibre optics has made possible would have seemed unnecessary as well as impossible. It was only in the seventies that people managed to exceed unboosted fibre optic transmission over a kilometre.

You speak of otherwise scarce wireline and fibre bandwidth, but they are not scarce. Lines do not interfere with each other. Any time your fibre gets congested, you simply lay ten more fibres, or ten thousand if you like. The main bandwidth problems at present involve passing the signal over long distances through air or space, not through cables.

That is why, as you put it, there is a common refrain that with all the new bandwidth hungry devices, and the mobile market expanding rapidly, that there is “a shortage of spectrum”.  As you say, this is obvious, and is obvious because the principles are by now very well understood, not because it is 'repeated often enough that it “must be true”.'   It is much like it is "obvious" that we cannot get a Type 1 perpetual motion machine because of the laws of thermodynamics. Anyone who thinks we can is welcome to try, but please not to waste our time with it before he has a working device, at which time he is welcome to a kilogram of Nobel medals.

You ask about forward looking science going on around these possibilities?  Certainly. We are talking big money here. LTE, since you ask, is more about the bread-and-butter protocols for applying and interconnecting  the technology than "forward looking science".  For the foreseeable future the main prospects are technological rather than exploratory science. 

The total bandwidth available to mobile devices and services is limited. In saying that I am recognising that only a limited number of bands are available. We do need spectrum for other services such as navigation, radar TV. microwave links, satellite systems and so on. Also some bands are not suitable because of their propagation characteristics and available bandwidths within the particular band.

In the 1940's Claude Shannon developed a theorem concerning the amount of information that could be transmitted over a given channel. The amount of information is determined by the bandwidth and the signal to noise ratio (noise includes interference from other channels). The greater the bandwidth the greater the capacity and the better the signal to noise ratio the greater the capacity. For any channel there is a limit to the amount of information that can be carried which is known as the "Shannon Limit".

All modern digital mobile systems work almost at the Shannon limit.

Banwidth is reused in cell systems by using any given channel or group of channels in multiple cells spaced so as to limit inbterference between them. Channels can be defined in the frequency domain, time domain or code domain. For explanations of these search for FDMA, TDMA and CDMA.

There is a system called BLAST that allows us to decode the strongest signal on a channel, subtract this from the total incomming signal so that we can see the next strongest which we decode and subtract from the remainder allowing us to detect and decode the next strongest and so on. This allows us to break the Snannon limit. This however requires a great deal of computing power and is difficult to implement in a realworld mobile environment.

So, at the moment, we are stuck with the Shannon limit. We can increase traffic carrying capacity by making our cells in cell networks smaller, but as you might imagine there are practical and financial limitations on this.

At one point AT&T "oversold the market" here in NYC. Problems placing cell phone calls and dropped calls at peak times resulted.

More capacity was added. All was fine but now with the internet on cell phones it is common to have problems at peak times again.

Dropped calls, calls that break down into data stream digital noise etc.

I think this will be a recurring issue.

With all the analog TV channels shut down there's new bandwidth to be divvied up.

an addendum primarily to stewarth's answer.

Cell phone techniquology has a clever, (sneaky) trick that I will attempt to elucidate using a simple linear town (one built along a road) as an example. Assume that we have one channel (a set frequency spectrum) whose bandwidth copes with 100 users and we set this tower one quarter way into this town of 100 people who all use their phone continuously.  This tower/100 cell phones combination communicates at full power, everything works hunky dory (except cell phone batteries drain quickly.)

Another 100 people move in so another cell tower is set up 3/4 way into town and this operates on a different channel also at full power. Things are still good!

Now (dammit!) 100 more nasty "illegal" immigrants (sarcasm) move in and we only have the 2 channels available. What do we do?

Simply install a tower in the middle of the town set to a channel (A) (to service only those in the middle of the town) and make sure the outer channels (serving only those in that particular outskirt) use channel B. Ahh! What about interference between these outside channels.

This is solved by reducing the power of those outer towers so they do not "spill" over each other's area.

In real life what happens is.

Every now and then each tower shouts out (full power and all channels - or a dedicated "management channel") "WHO CAN HEAR ME? AND HOW LOUDLY DO YOU HEAR ME?" and all the cell phones that are on (not necessarily "being used") reply and the tower now knows how faraway each one is. Now the towers arbitrate amongst themselves who will manage which cell phones (the closest), which channel to use and they will use the least possible power required to service their cell phones, This solves cross talk of nearby (but not adjacent) towers (and adjacent towers use different channels to solve that problem)

In our example the middle tower could use full power and the outer towers on the same channel reduce their power so that their signals do not extend into each others areas.

More people move in so we simply put a tower on one or both extemities of our linear town using the same channel as the midtown tower but all three towers reduce their power to only serve their area.

With the addition of a few more channels this technique can be extended (infinitely) to a 2 dimensional town. Communication between towers is by hardwire or very directional dish microwave which prevents unwanted spillover.

As more users need to be serviced, it is simply a case of more towers (that operate at lower powers) being installed.

(Most!) Talking humans automatically do this when a "town hall meeting/party" is convened and matters are discussed in groups of different shifting sizes. The presently speaking persons adjusts the volumes of their voices to allow everyone in their group to hear, but which doesn't interfere with the neighbouring group.

This technique is one of those "I wish I'd thought of that" ideas!