It doesn't quite work this way. This is going to be a bit technical, but you asked a technical question, so bear with me. Yes, I am a ham (since you asked for one), and I've also done some commercial RF data systems.
As others have pointed out, cellular telephone systems aren't like broadcast systems. You really can "put up more towers" to increase the amount of "service" (available data transfer per unit time, number of simultaneous voice calls, etc.) in a given geographic area without using more RF bandwidth. The reason for this is that you can turn the power on the base and handset down to reduce the coverage of the cell allowing reuse of the RF bandwidth more frequently within a certain geographical space. This is already done: cells on rural highways are much larger than cells within a city. In fact, the cells on rural highways would often be capable of covering an entire city from a geographic point of view, but there wouldn't be enough capacity to handle all that traffic, so smaller (lower power, lower antenna angle, etc.) cells are placed in cities allowing reuse of that RF bandwidth. Broadcast services can be thought of as "cellular" with very large cells (depending on the service, up to and including the entire planet for HF "shortwave" radio, for example) if you want, but that's not a traditional interpretation.
As for how much bandwidth it takes to attain a certain information rate, that varies with a number of factors. Assuming a uniform RF environment (noise, propagation, etc., which of course isn't true but is handy for discussion), the key tradeoff is made by how "aggressive" your modulation scheme is. A more aggressive modulation scheme packs more data into a certain amount of RF bandwidth, but it requires a stronger signal to noise ratio at the receiver to demodulate and recover the data. The exact relationship between how much data you can chuck into a given amount of RF bandwidth and the required receiver SNR varies with your chosen modulation scheme and receiver design. The reason data rates have been increasing with time is that newer, better (easier to demodulation) modulation schemes and better (mostly less noisy, but also more cost effective for a given complexity) receivers are being developed. More cells are also being added (see above) to lessen "competition" for the channel's bandwidth, but we're also seeing a lot more users and demand, so that probably averages out. The amount of RF bandwidth allocated to the cellular telephone services has remained roughly constant since the late 90s (800MHz cellular band + 1900MHz PCS band, though other bands are also used regionally, and some of these are new).
In a two-way scenario like a cellular telephone, you also get to play with the fact that the two directions don't behave equally. The base-to-handset link (downlink) has the advantage of no access contention (there's just one base, and it knows everything it's doing), expensive equipment (there's only one, so the company can pump some money into it), and lots of power available (it's plugged into the wall). The handset-to-base link (uplink) is messier: it has access contention (multiple handsets coordinated remotely by the base), cost sensitive equipment (consumers don't like to pay thousands of dollars for their handsets), and limited power (batteries). Antennas are something of a wash since antennas are effective about equally in both directions. What all this means is that it's easier to use a more aggressive modulation scheme (and hence cram more bits per second into a given chunk of RF MHz) on the downlink than the uplink. Fortunately, this is roughly in-line with consumer demand: most consumers want to transfer large stuff to their phones, not from them. FWIW, Cable Modems have similar concerns, and a similar situation results.
You also seem to assume a TDMA based uplink channel. Modern standards are all CDMA based. While the theory of operation is totally different, the effect is the same: multiple people contend for the same resource. Various standards allow for various amounts of "resource concatenation" to allow one user to use more than one "unit" of the uplink when it's not full, but most of them do not dynamically adjust the size of the quantum. (Though, somewhat amusingly, the downlink of CDMA2000 EVDO is time-division multiplexed).
Cell providers also have to ensure that old handsets are supported. For non-compatible upgrades (e.g. GSM to UMTS - done by AT&T and T-Mobile), this means they basically have to run both standards side-by-side on different channels. This results in some interesting scenarios where congestion will force some users onto a slower, less efficient standard even if the handset and base both support something better. I've seen this happen at amusement parks a lot.
Also, some on-air standards don't support voice calls at all (e.g. EVDO or HSDPA), and the carriers always want to be able to handle voice calls, so generally at least one channel on a cell will operate in a voice capable mode, though I can imagine some might have the capability to drop every channel into a data-only mode if there really are no voice calls. A true "4G" service would eliminate this as voice calls are actually just data, as far as the infrastructure is concerned, with QoS used to ensure availability. Current data standards (including the LTE being deployed by Verizon) do not have this QoS capability, so voice calls are still routed on a separate channel (probably RTT, but I'm not familiar enough with these deployments to say this authoritatively). LTE Advanced should mash all this back together.
Of course, all this stuff has limits. It's not practical to set up a cell (with the required base station, antennas, tower, etc.) every 100 feet, and current RF technology and transmitter/receiver complexity is only so advanced, so there is a limited amount of service available to an end user. I can't say I agree with Verizon's pricing (I use Sprint, and while I don't have a 4G handset, if I did, I'd have unlimited transfer while on 4G Wi-Max and a 5GB soft cap while on 3G EVDO/RTT), it's not like they could offer unlimited everything easily. I just think the overage charges and base limits are a little off.
(I apologize if I've gotten any technical details e.g. what standard supports what wrong - I'm not a cellular engineer, so if anyone reading this is, please correct me. The basics should be reasonable, though)