Nothing Special
There is nothing particularly special about ISO 1600, although in some cases ISO settings beyond 1600 have incurred less effective and efficient ways of amplifying the image signal. When you set ISO on a camera, that is simply instructing the camera to change the maximum saturation point of the sensor, from which the signal will be amplified. Increasing ISO will usually instruct the camera to amplify the signal before downstream electronics add more electronic and quantization noise, so using a higher in-camera ISO is usually (with caveats) better than using a low ISO and boosting exposure in post when you can't get enough light down the lens. There is no special math behind ISO 1600 specifically being the "best" high ISO setting to minimize noise in all cases, however there may be brand-specific amplification mechanics that affect the quality of noise at high ISO in some cases. Noise is a function of two key factors: electronic noise present in the circuit of the sensor & the lesser contributor of noise, and photon shot noise, sometimes called Gaussian noise, which is the primary contributor of noise. How noise exhibits (noise quality) is a factor of the method(s) of amplification.
Noise
Noise is an attribute of the image signal, and ultimately has to do with how the sensor is designed, the nature of its fabrication, signal to noise ratio (S/N), signal gain, ADC efficiency, and a number of other factors. These factors differ from brand to brand, model to model, and in some ways even from one camera to the next of the same model. Newer sensors generally tend to exhibit less noise in general than older sensors, regardless of pixel size or ISO setting used. The highest usable ISO setting can differ from two samples of the same camera (which is often touted as the case with Canon 7D DSLRs) and from cameras of differing generations (such as the Canon 400D and 650D).
Electronic noise, which can have a variety of forms such as Fixed Pattern Noise (FPN), Horizontal and Vertical Banding Noise (HVBN), is often unnaturally patterned and thus very undesirable, but only affects the lowest levels of the image signal (i.e. the deep shadows). As the ISO setting is increased, electronic noise exhibits less and less, and is ultimately entirely overpowered by photon noise.
Photon noise is the result of the random nature of light, which follows a Poisson distribution at the sensor. That means photons randomly strike the sensor, but at a high enough signal level their distribution is roughly even, and thus photon noise exhibits in the same way at any location on the sensor. Photo noise accounts for the very vast majority of noise in a digital photograph, orders of magnitude more than electronic noise (with the exception of improperly exposed photos where the signal is only a tiny fraction of the full well capacity, or FWC.)
There are several factors that affect how significantly photon noise will exhibit in a photo. Historically, the larger the pixel, the less this type of noise would exhibit. A photodiode is sensitive to light over area...photon penetration depth into the silicon of the photodiode is not a factor of pixel saturation. Larger pixels usually allow the use of higher ISO settings, as they capture more photons per unit time. More photons per unit time means more photons overall for any given exposure, which increases S/N. A higher S/N allows the use of a higher gain (more on this in a moment) when converting the analog sensor signal into a digital signal (RAW file), which is what your ultimately use to generate JPEGs and TIFF images with post processing software.
Gain
Gain is the conversion ratio of electrons (e-) to digital units (DU). A camera that converts exactly one e- to one DU has "unity gain". Most cameras achieve unity gain at some exact (but possibly non-selectable) ISO setting. More frequently, gain is fractional, such as 5.7 e- to every DU. For every stop increase in ISO, gain drops by the same factor. If you have a gain of 5.7 e-/DU at ISO 100, you would have 2.85 e-/DU at ISO 200, 1.425 e-/DU at ISO 400, .7125 e-/DU at ISO 800, and 0.35625 e-/DU at ISO 1600. As you increase ISO, you lose signal to noise ratio (S/N). A lower S/N is never really a good thing...it always means more noise due to a lesser signal being amplified. A lesser signal means less color fidelity along with less detail. The point at which those facts cause an image to degrade ultimately depends on the mechanism of amplification, which can differ from generation to generation and brand to brand.
Newer sensor technology has been increasing full well capacity generation over generation, even as pixel area shrinks by employing more effective means of directing photons to a light-sensitive surface rather than light absorbing surfaces and components. The relatively recent introduction of microlenses to CMOS sensors has helped direct photons onto the sensitive surface of the photodiode, and away from readout wiring and other non-sensitive surfaces. Lightpipe technology uses high refractive index material below specially tuned microlenses to help guide light through the channel of readout wiring above the photodiode, such that more of it reaches the diode instead of reflecting off the wiring. Backside illuminated sensors simply flip the whole structure, exposing the photodiode directly to light, eliminating the need for all of the other structures. All of these things improve "quantum efficiency" (Q.E.), or the overall conversion rate of photons to electrons in the sensor. Sensors with a higher Q.E. support higher maximum signal to noise ratios, which in turn support higher ISO 100 gain...which further supports higher gain at lower ISO. Higher gain, more electrons per digital unit, which lessens the effects of photon noise at every ISO setting.
Mechanisms of Amplification
Now onto the reasons why ISO 1600 (in the past) was the highest "useful" ISO setting in many cases. Some brands, such as Canon and in some cases Nikon, use more than one mechanism to amplify the signal coming off the sensor. In the last several years, ISO 1600 was usually the last "natively amplified" ISO setting, beyond which additional amplifiers or even digital amplification was used to achieve the next ISO setting. Canon is probably the worst offender to use alternative amplification mechanisms. In their last generation of cameras (7D, 5D II, 1D/s III, and all of the Rebel series up to the 650D) used an additional analog gain on the pixel stream...after pixel read but before ADC (analog-to-digital conversion), to achieve ISO settings above ISO 1600. Nikon used something similar in sensors they manufactured themselves (any Nikon cameras, and for that matter any other brands, that use Sony Exmor sensors use a radically different approach to signal management in general, so they don't apply here.)
Achieving ISO 3200 in the past would employ standard per-pixel analog gain at the time of pixel readout for all full-stop ISO settings up through ISO 1600, then an additional analog gain to the pixel stream coming off the sensor. In some cameras, ISO 6400 would use the same alternative analog gain post-read. ISO settings above ISO 6400 usually used a metadata digital gain hint to instruct post processing tools to apply an additional digital gain to achieve higher ISO settings. Such settings were usually called "Expanded" or "High" ISO settings, and could only be used in full stop increments above the cameras "native" ISO setting. (Note: The only reason one would ever really need to use an expanded ISO setting in-camera would be if they absolutely required a higher shutter speed than could be achieved with a lower native ISO setting. In many cases, still choosing the lower ISO setting at the required shutter speed, which would under expose, could still be desirable as manually fixing the exposure in post will usually produce a better result than in-camera expanded ISO.)
The information above can be applied to cameras from the last couple generations, not including the current generation. Older cameras from Canon definitely employ native analog gain as well as additional post-read analog gain, as well as a potential hard-coded (i.e. non-configurable) +/- 1/3 stop push or pull executed by the camera via behind-the-scenes exposure adjustments. This push/pull tended to cost you a loss of about 1/3rd stop of dynamic range. Past generation Nikon cameras that used sensors designed by Nikon also employed similar high ISO gain, although they seem to use analog gain for all ISO settings (including third-stops), which usually resulted in better IQ at high ISO third stop settings relative to Canon, without any DR loss. The current generation of cameras from Canon seem to employ a better analog gain approach to ISO settings up to the new maximums (25600 in the case of their non-1D lines, and 51200 in the case of the 1D X), with the additional post-read gain only being employed for the highest ISO setting (at least, as seems to be indicated by testing of these cameras so far).
That would mean that the highest usable ISO for Canon cameras has jumped from ISO 1600 to ISO 12800 at least, and possibly even ISO 25600 for the 1D X. Sony Exmor sensors, which are now used by the majority of Canon's competition including Nikon, employ a very different kind of sensor design and processing architecture. Exmor sensors top out at a native ISO 12800, and all further ISO settings are expanded mode ISO. Up through ISO 12800, Sony Exmor sensors perform quite well, on par with the 5D III and 1D X. Beyond that, noise quality starts to fall apart pretty quickly, and doesn't generally hold up to Canon ISOs 16000, 20000, 25600, 32000, 40000, and 51200. On the flip side, Sony Exmor has barely any read noise, and performs considerably better at ISO 100, 200, and to some degree even 400 in terms of dynamic range. Exmor DR has quickly become a thing of legend, and the sensor really shines for types of photography that eat dynamic range for breakfast (such as Landscapes).
Usable ISO 3200...6400...16000?
The constant stream of new and improved technology is constantly changing things. Barely four years ago, the Canon 450D and 40D could barely do ISO 800, with ISO 1600 being largely unusable. A generation later, ISO 1600 became more usable, and in the case of the 5D II and 1Ds III, ISO 3200 was even "usable" in some circumstances. Today, I regularly hear, particularly from sports photographers and photojournalists, that ISO up to 16000, 20000, and at times even 25600 on the 1D X is "entirely usable", "even printable with some post-processing work!" From a mathematic standpoint, electrons and digital units and gain and all that, I wouldn't necessarily say there is anything in particular that calls out ISO 1600 as the magic ISO number. The highest usable ISO has been increasing generation over generation, usually by around one stop, but recently with Canon's newest sensors, it has increased by as much as three, possibly even four stops.
